Proof of The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree

Each new version treated here has the text of its earlier version, except for the following changes; every other reference, the settings included, is the same. References to Sequences, Convergent Sequences of Real Numbers and The Limit of a Convergent Sequence point to Sequences, Convergent Sequences of Real Numbers and The Limit of a Convergent Sequence. In Sequences, Convergent Sequences of Real Numbers, Cauchy Sequences of Real Numbers and Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence, the index set is named as the set of natural numbers, which is the set $\mathbb{N}$ of The Set of Natural Numbers and the Number One (i) by The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (i). In Convergent Sequences of Real Numbers, The Limit of a Convergent Sequence, Cauchy Sequences of Real Numbers and Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence, the words "real numbers", or "real number", carry a link to The Real Numbers as a Type; an element of $\mathrm{Real}$ in that sense is given by its set, an element of $\mathbb{R}$ (The Real Numbers as a Type (i)), and corresponds to the element of the earlier $\mathrm{Real}$ with the same set by The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (v), so the real numbers meant are still the elements of $\mathbb{R}$. Finally, Convergent Sequences of Real Numbers states convergence to $L$, being convergent and being a null sequence in three clauses where the earlier version uses one, and Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii) adds that the sum, the limit of $(s_{n})$, is the real number to which $(s_{n})$ converges, which is what The Limit of a Convergent Sequence (i) says. So each condition is the same condition once the notions it cites are known to agree.

(i): in both versions a sequence in $X$ is a map $a:\mathbb{N}\to X$ with $n$-th term $a(n)$, with the same $\mathbb{N}$.

(ii): by (i) both versions speak of the same sequences in $\mathbb{R}$, and both require that for every real $\varepsilon>0$ there be $N\in\mathbb{N}$ with $|a_{n}-L|<\varepsilon$ for all $n\ge N$. So convergence to $L$ agrees, and with it being convergent and being a null sequence.

(iii): a sequence is convergent in either sense, by (ii), and both versions define the limit as the unique real number to which the sequence converges, which is the same number by (ii).

(iv): by (i) both versions speak of the same sequences, and the condition on $|a_{m}-a_{n}|$ is the same.

(v): in both versions the partial sums (Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (i)) are the finite sums of Sums and Products over a Finite Set and over an Interval (v), and for every sequence $(a_{k})$ in $\mathbb{R}$ the sequence $(s_{n})$ they form and the sequence $(|a_{k}|)$ exist (Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (i), Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (iv)). The series converges when $(s_{n})$ is convergent, which agrees by (ii); its sum is the limit of $(s_{n})$, which agrees by (iii); and absolute convergence is convergence of the series of the $|a_{k}|$, which agrees by what was just shown.

(vi): Iterated Operations: Finite Sums and Finite Products is Iterated Operations: Finite Sums and Finite Products word for word, with the same references, so iterated operations, their restricted form, and finite sums and products coincide.