Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let convergence and null sequences be as in Convergent Sequences of Real Numbers §converges, bounded sequences as in Bounded Sequences of Real Numbers §bounded and subsequences as in Subsequences §subsequence. Let $(a_{n})$, $(b_{n})$ and $(c_{n})$ be sequences in $\mathbb{R}$, and let $A,B,\lambda,L\in\mathbb{R}$ and $K\in\mathbb{N}$. In the clauses tails, arithmetic, quotient, absolute, order, squeeze and subsequence, assume $a_{n}\to A$, and where $B$ occurs also $b_{n}\to B$.

¶ The constant sequence $(\lambda)_{n\in\mathbb{N}}$ converges to $\lambda$.

¶ If $c_{n}\to L$ and $c_{n}\to L'$ with $L'\in\mathbb{R}$, then $L=L'$.

¶ Every convergent sequence in $\mathbb{R}$ is bounded.

¶ A sequence in $\mathbb{R}$ is bounded if and only if it is bounded above and bounded below.

¶ If $c_{n}=a_{n}$ for every $n\ge K$, then $c_{n}\to A$; and $(a_{n+p})_{n\in\mathbb{N}}$ converges to $A$ for every $p\in\mathbb{N}$.

¶ $a_{n}+b_{n}\to A+B$, $\lambda a_{n}\to\lambda A$, $-a_{n}\to-A$, $a_{n}-b_{n}\to A-B$ and $a_{n}b_{n}\to AB$.

¶ If $B\neq0$ and $b_{n}\neq0$ for every $n\in\mathbb{N}$, then $a_{n}/b_{n}\to A/B$.

¶ $|a_{n}|\to|A|$.

¶ Let $I$ be a finite set, $d:I\times\mathbb{N}\to\mathbb{R}$ and $D:I\to\mathbb{R}$, $i\mapsto D_{i}$, maps such that $(d(i,n))_{n\in\mathbb{N}}$ converges to $D_{i}$ for every $i\in I$. Then $\big(\sum_{i\in I}d(i,n)\big)_{n\in\mathbb{N}}$ converges to $\sum_{i\in I}D_{i}$.

¶ If $a_{n}\le b_{n}$ for every $n\ge K$, then $A\le B$. If $a_{n}\le\lambda$ for every $n\ge K$, then $A\le\lambda$, and if $\lambda\le a_{n}$ for every $n\ge K$, then $\lambda\le A$.

¶ If $A=B$ and $a_{n}\le c_{n}\le b_{n}$ for every $n\ge K$, then $c_{n}\to A$.

¶ If $(b_{n})$ is a null sequence and $|c_{n}-L|\le b_{n}$ for every $n\ge K$, then $c_{n}\to L$.

¶ Every subsequence of $(a_{n})$ converges to $A$.