Proof of Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence

Each result cited is universally quantified over the data in its own statement.

Notation. For a bounded sequence $(c_{n})$ in $\mathbb{R}$ (below, $(a_{n})$, $(b_{n})$ or $(-a_{n})$) and $k\in\mathbb{N}$, let $T_{k}(c)=\{c_{m}:m\in\mathbb{N},\ m\ge k\}$. By Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §tails, $T_{k}(c)$ is nonempty and bounded and has a supremum $\overline{c}_{k}$ and an infimum $\underline{c}_{k}$, unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum; by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §sandwich, $\underline{c}_{k}\le c_{k}\le\overline{c}_{k}$; and by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §monotone, $(\overline{c}_{k})_{k\in\mathbb{N}}$ is nonincreasing and $(\underline{c}_{k})_{k\in\mathbb{N}}$ is nondecreasing. By Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum and Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least, $\overline{c}_{k}$ is an upper bound of $T_{k}(c)$ that is $\le$ every upper bound of $T_{k}(c)$, and $\underline{c}_{k}$ is a lower bound of $T_{k}(c)$ that is $\ge$ every lower bound of $T_{k}(c)$, bounds being as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §bounds; call this (E). By The Limit Superior and the Limit Inferior of a Bounded Sequence of Real Numbers §limsup, The Limit Superior and the Limit Inferior of a Bounded Sequence of Real Numbers §liminf and The Limit of a Convergent Sequence §limit, $\overline{c}_{k}\to\limsup_{n\to\infty}c_{n}$ and $\underline{c}_{k}\to\liminf_{n\to\infty}c_{n}$ as $k\to\infty$; call this (L). Write $\overline{\alpha}=\limsup_{n\to\infty}a_{n}$, $\underline{\alpha}=\liminf_{n\to\infty}a_{n}$, $\overline{\beta}=\limsup_{n\to\infty}b_{n}$ and $\underline{\beta}=\liminf_{n\to\infty}b_{n}$. The order of $\mathbb{N}$ is transitive by Arithmetic and Order of the Natural Numbers §partial-order.

Clause formula. By (L), $\overline{a}_{k}\to\overline{\alpha}$ and $\underline{a}_{k}\to\underline{\alpha}$; by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §limits, $\inf\{\overline{a}_{k}:k\in\mathbb{N}\}$ and $\sup\{\underline{a}_{k}:k\in\mathbb{N}\}$ exist and $\overline{a}_{k}\to\inf\{\overline{a}_{k}:k\in\mathbb{N}\}$ and $\underline{a}_{k}\to\sup\{\underline{a}_{k}:k\in\mathbb{N}\}$; so $\overline{\alpha}=\inf\{\overline{a}_{k}:k\in\mathbb{N}\}$ and $\underline{\alpha}=\sup\{\underline{a}_{k}:k\in\mathbb{N}\}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §unique.

Clause order. By (L), $\underline{a}_{k}\to\underline{\alpha}$ and $\overline{a}_{k}\to\overline{\alpha}$, and $\underline{a}_{k}\le a_{k}\le\overline{a}_{k}$, so $\underline{a}_{k}\le\overline{a}_{k}$, for every $k\in\mathbb{N}$ by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §sandwich. The first sentence of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order, applied to $(\underline{a}_{k})$ and $(\overline{a}_{k})$ with $K=1$, gives $\underline{\alpha}\le\overline{\alpha}$.

Clause comparison. Assume $a_{n}\le b_{n}$ for every $n\ge K$, and let $k\in\mathbb{N}$ with $k\ge K$. Every element of $T_{k}(a)$ is $a_{m}$ for some $m\ge k$, hence $m\ge K$, so $a_{m}\le b_{m}\le\overline{b}_{k}$ by (E) for $(b_{n})$; thus $\overline{b}_{k}$ is an upper bound of $T_{k}(a)$ and $\overline{a}_{k}\le\overline{b}_{k}$ by (E) for $(a_{n})$. Likewise every element $b_{m}$ of $T_{k}(b)$, with $m\ge k\ge K$, satisfies $\underline{a}_{k}\le a_{m}\le b_{m}$, so $\underline{a}_{k}$ is a lower bound of $T_{k}(b)$ and $\underline{a}_{k}\le\underline{b}_{k}$ by (E). By (L), $\overline{a}_{k}\to\overline{\alpha}$, $\overline{b}_{k}\to\overline{\beta}$, $\underline{a}_{k}\to\underline{\alpha}$ and $\underline{b}_{k}\to\underline{\beta}$, so the first sentence of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order, with this $K$, gives $\overline{\alpha}\le\overline{\beta}$ and $\underline{\alpha}\le\underline{\beta}$.

Clause bounds. Assume $a_{n}\le M$ for every $n\ge K$. For $k\ge K$, every element $a_{m}$ of $T_{k}(a)$ has $m\ge k\ge K$, so $M$ is an upper bound of $T_{k}(a)$ and $\overline{a}_{k}\le M$ by (E). As $\overline{a}_{k}\to\overline{\alpha}$ by (L), the second sentence of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order, with the constant there taken to be $M$, gives $\overline{\alpha}\le M$. If instead $M\le a_{n}$ for every $n\ge K$, then in the same way $M$ is a lower bound of $T_{k}(a)$ for $k\ge K$, so $M\le\underline{a}_{k}$ by (E), and $M\le\underline{\alpha}$ by (L) and the last part of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order.

Clause eventually. Let $\varepsilon$ be a real number with $\varepsilon>0$. The choices are made in this order. First, Convergent Sequences of Real Numbers §converges, applied to $\overline{a}_{k}\to\overline{\alpha}$ from (L) and this $\varepsilon$, gives $N_{1}\in\mathbb{N}$ with $|\overline{a}_{k}-\overline{\alpha}|<\varepsilon$ for every $k\ge N_{1}$. Second, the same definition applied to $\underline{a}_{k}\to\underline{\alpha}$ and $\varepsilon$ gives $N_{2}\in\mathbb{N}$ with $|\underline{a}_{k}-\underline{\alpha}|<\varepsilon$ for every $k\ge N_{2}$. Third, let $N=\max\{N_{1},N_{2}\}$, the greatest element of $\{N_{1},N_{2}\}$, which exists by The Maximum and Minimum of Two Elements of a Total Order §exists and satisfies $N_{1}\le N$ and $N_{2}\le N$ by The Maximum and Minimum of Two Elements of a Total Order §bounds. Taking $k=N_{1}$, respectively $k=N_{2}$, the ordered-field rules give $\overline{a}_{N_{1}}<\overline{\alpha}+\varepsilon$ and $\underline{\alpha}-\varepsilon<\underline{a}_{N_{2}}$. Now let $m\in\mathbb{N}$ with $m\ge N$. Then $m\ge N_{1}$ and $m\ge N_{2}$, so $a_{m}\in T_{N_{1}}(a)$ and $a_{m}\in T_{N_{2}}(a)$, and by (E), $\underline{\alpha}-\varepsilon<\underline{a}_{N_{2}}\le a_{m}\le\overline{a}_{N_{1}}<\overline{\alpha}+\varepsilon$.

Clause frequently. Let $\varepsilon$ be a real number with $\varepsilon>0$, and let $N\in\mathbb{N}$. As $(\overline{a}_{k})$ is nonincreasing, $\overline{a}_{k}\le\overline{a}_{N}$ for every $k\ge N$ by Monotone Sequences and Subsequences: Comparison of All Terms, Growth of the Indices, and Subsequences of Subsequences §monotone, so $\overline{\alpha}\le\overline{a}_{N}$ by (L) and the second sentence of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order, with the constant there taken to be $\overline{a}_{N}$ and $K=N$. Hence $\overline{\alpha}-\varepsilon<\overline{a}_{N}=\sup T_{N}(a)$, and since $T_{N}(a)$ is nonempty and bounded above, Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers §strict-above, applied to $S=T_{N}(a)$ and $b=\overline{\alpha}-\varepsilon$, gives an element of $T_{N}(a)$ greater than $\overline{\alpha}-\varepsilon$, that is, $m\in\mathbb{N}$ with $m\ge N$ and $\overline{\alpha}-\varepsilon<a_{m}$. Dually, as $(\underline{a}_{k})$ is nondecreasing, $\underline{a}_{N}\le\underline{a}_{k}$ for every $k\ge N$ by Monotone Sequences and Subsequences: Comparison of All Terms, Growth of the Indices, and Subsequences of Subsequences §monotone, so $\underline{a}_{N}\le\underline{\alpha}$ by (L) and the last part of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order; hence $\inf T_{N}(a)=\underline{a}_{N}<\underline{\alpha}+\varepsilon$, and Arbitrary Positive Slack, and Approximation of Suprema and Infima, in the Real Numbers §strict-below, applied to $S=T_{N}(a)$ and $b=\underline{\alpha}+\varepsilon$, gives $m'\in\mathbb{N}$ with $m'\ge N$ and $a_{m'}<\underline{\alpha}+\varepsilon$.

Clause subsequences. For $j\in\mathbb{N}$, $j+1\in\mathbb{N}$, so $0<j$ and $0<j+1$, and hence $0<1/j$ and $0<1/(j+1)$; and $u<u+1$ for $u\in\mathbb{N}$ by Arithmetic and Order of the Natural Numbers §successor. For $j,u\in\mathbb{N}$ let $S(j,u)=\{m\in\mathbb{N}:u<m,\ \overline{\alpha}-1/(j+1)<a_{m}\}$ and $S_{1}=\{m\in\mathbb{N}:\overline{\alpha}-1<a_{m}\}$. By clause frequently above, applied with $\varepsilon=1/(j+1)$ and $N=u+1$ (an element $m\ge u+1$ satisfies $u<m$), respectively with $\varepsilon=1$ and $N=1$, these subsets of $\mathbb{N}$ are nonempty, so each has a least element by Arithmetic and Order of the Natural Numbers §well-order, unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least. Let $c=\min S_{1}$ and let $g:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ be given by $g(j,u)=\min S(j,u)$, a map (a subclass of the set $(\mathbb{N}\times\mathbb{N})\times\mathbb{N}$) by Maps and Relations Given by Formulas §binary, applied with all three sets equal to $\mathbb{N}$ and the expression $\min S(j,u)$, which lies in $\mathbb{N}$; no choice is involved. By Recursion on the Natural Numbers Starting at One §recursion there is a map $\sigma:\mathbb{N}\to\mathbb{N}$ with $\sigma(1)=c$ and $\sigma(j+1)=g(j,\sigma(j))\in S(j,\sigma(j))$ for every $j\in\mathbb{N}$. Thus $\sigma(j)<\sigma(j+1)$ for every $j$, so $\sigma$ is strictly increasing by Monotone Sequences §monotone and $(a_{\sigma(j)})_{j\in\mathbb{N}}$ is a subsequence of $(a_{n})$ by Subsequences §subsequence; and $\overline{\alpha}-1/j<a_{\sigma(j)}$ for every $j\in\mathbb{N}$, since $j=1$ or $j=d+1$ with $d\in\mathbb{N}$ by Arithmetic and Order of the Natural Numbers §predecessor, and $\sigma(1)\in S_{1}$, $\sigma(d+1)\in S(d,\sigma(d))$. Also $a_{\sigma(j)}\le\overline{a}_{\sigma(j)}$ by Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §sandwich. Now $1/j\to0$ by Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences §reciprocals and the constant sequence $(\overline{\alpha})$ converges to $\overline{\alpha}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §constant, so $\overline{\alpha}-1/j\to\overline{\alpha}-0=\overline{\alpha}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic; and $(\overline{a}_{\sigma(j)})_{j\in\mathbb{N}}$ is a subsequence of $(\overline{a}_{k})$, which converges to $\overline{\alpha}$ by (L), so $\overline{a}_{\sigma(j)}\to\overline{\alpha}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §subsequence. Hence $a_{\sigma(j)}\to\overline{\alpha}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §squeeze with $K=1$. The same construction with $S(j,u)=\{m\in\mathbb{N}:u<m,\ a_{m}<\underline{\alpha}+1/(j+1)\}$ and $S_{1}=\{m\in\mathbb{N}:a_{m}<\underline{\alpha}+1\}$, nonempty by the second half of clause frequently above, gives a strictly increasing $\tau:\mathbb{N}\to\mathbb{N}$ with $\underline{a}_{\tau(j)}\le a_{\tau(j)}<\underline{\alpha}+1/j$ for every $j\in\mathbb{N}$, and $a_{\tau(j)}\to\underline{\alpha}$ in the same way, using $\underline{\alpha}+1/j\to\underline{\alpha}$ and $\underline{a}_{\tau(j)}\to\underline{\alpha}$.

Clause subsequence-limits. Let $\rho:\mathbb{N}\to\mathbb{N}$ be strictly increasing with $a_{\rho(j)}\to L$, as Subsequences §subsequence allows. By Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §sandwich, $\underline{a}_{\rho(j)}\le a_{\rho(j)}\le\overline{a}_{\rho(j)}$ for every $j\in\mathbb{N}$; by (L) and Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §subsequence, $\underline{a}_{\rho(j)}\to\underline{\alpha}$ and $\overline{a}_{\rho(j)}\to\overline{\alpha}$; so $\underline{\alpha}\le L\le\overline{\alpha}$ by the first sentence of Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §order with $K=1$.

Clause extraction. Let $\varepsilon$ be a real number with $\varepsilon>0$. For $u\in\mathbb{N}$ let $P(u)=\{m\in\mathbb{N}:u<m,\ \overline{\alpha}-\varepsilon<a_{m}\}$, and let $P_{1}=\{m\in\mathbb{N}:\overline{\alpha}-\varepsilon<a_{m}\}$. By clause frequently above, applied with this $\varepsilon$ and $N=u+1$ (an element $m\ge u+1$ satisfies $u<m$, as $u<u+1$), respectively with this $\varepsilon$ and $N=1$, these subsets of $\mathbb{N}$ are nonempty, so each has a least element by Arithmetic and Order of the Natural Numbers §well-order, unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least. Let $h:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ be given by $h(j,u)=\min P(u)$, a map by Maps and Relations Given by Formulas §binary as for $g$ in clause subsequences above. By Recursion on the Natural Numbers Starting at One §recursion there is a map $\sigma:\mathbb{N}\to\mathbb{N}$ with $\sigma(1)=\min P_{1}$ and $\sigma(j+1)=h(j,\sigma(j))\in P(\sigma(j))$ for every $j\in\mathbb{N}$. Thus $\sigma(j)<\sigma(j+1)$ for every $j$, so $\sigma$ is strictly increasing by Monotone Sequences §monotone; and $\overline{\alpha}-\varepsilon<a_{\sigma(j)}$ for every $j\in\mathbb{N}$, since $j=1$ or $j=d+1$ with $d\in\mathbb{N}$ by Arithmetic and Order of the Natural Numbers §predecessor, and $\sigma(1)\in P_{1}$, $\sigma(d+1)\in P(\sigma(d))$. The same construction with $P(u)=\{m\in\mathbb{N}:u<m,\ a_{m}<\underline{\alpha}+\varepsilon\}$ and $P_{1}=\{m\in\mathbb{N}:a_{m}<\underline{\alpha}+\varepsilon\}$, nonempty by the second half of clause frequently above, gives a strictly increasing $\tau:\mathbb{N}\to\mathbb{N}$ with $a_{\tau(j)}<\underline{\alpha}+\varepsilon$ for every $j\in\mathbb{N}$.

Clause negation. By Bounded Sequences of Real Numbers §bounded there is $M_{0}\in\mathbb{R}$ with $|a_{n}|\le M_{0}$ for every $n\in\mathbb{N}$; as $|-a_{n}|=|a_{n}|$ by the ordered-field rules, $(-a_{n})$ is bounded by the same definition. Fix $k\in\mathbb{N}$; the elements of $T_{k}(-a)$ are the $-a_{m}$ with $m\ge k$. For such $m$, $\underline{a}_{k}\le a_{m}$ by (E), so $-a_{m}\le-\underline{a}_{k}$, and $-\underline{a}_{k}$ is an upper bound of $T_{k}(-a)$. If $v$ is any upper bound of $T_{k}(-a)$, then $-a_{m}\le v$, that is $-v\le a_{m}$, for every $m\ge k$, so $-v$ is a lower bound of $T_{k}(a)$, $-v\le\underline{a}_{k}$ by (E), and $-\underline{a}_{k}\le v$. Thus $-\underline{a}_{k}$ is the least upper bound of $T_{k}(-a)$, and by the uniqueness in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum, $\overline{(-a)}_{k}=-\underline{a}_{k}$. Exchanging upper and lower bounds in this argument gives $\underline{(-a)}_{k}=-\overline{a}_{k}$. By (L) and Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic, $\overline{(-a)}_{k}=-\underline{a}_{k}\to-\underline{\alpha}$, while $\overline{(-a)}_{k}\to\limsup_{n\to\infty}(-a_{n})$ by (L); so $\limsup_{n\to\infty}(-a_{n})=-\underline{\alpha}$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §unique. Likewise $\underline{(-a)}_{k}=-\overline{a}_{k}\to-\overline{\alpha}$ and $\underline{(-a)}_{k}\to\liminf_{n\to\infty}(-a_{n})$, so $\liminf_{n\to\infty}(-a_{n})=-\overline{\alpha}$.

Clause convergence. Assume $a_{n}\to A$. By clause subsequences above there are strictly increasing $\sigma,\tau:\mathbb{N}\to\mathbb{N}$ with $a_{\sigma(j)}\to\overline{\alpha}$ and $a_{\tau(j)}\to\underline{\alpha}$; by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §subsequence both subsequences also converge to $A$, so $\overline{\alpha}=A$ and $\underline{\alpha}=A$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §unique. Conversely, assume $\underline{\alpha}=A=\overline{\alpha}$. By Tail Suprema and Tail Infima of a Bounded Sequence of Real Numbers §sandwich, $\underline{a}_{n}\le a_{n}\le\overline{a}_{n}$ for every $n\in\mathbb{N}$, and by (L), $\underline{a}_{n}\to A$ and $\overline{a}_{n}\to A$; so $a_{n}\to A$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §squeeze with $K=1$.