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

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness.

(i) Sequences. For every set $X$, a map $a$ is a sequence in $X$ in the earlier version if and only if it is a sequence in $X$ in the new version, and it has the same terms in both.

(ii) Convergent sequences. For every sequence $(a_{n})$ in $\mathbb{R}$ and every $L\in\mathbb{R}$, $(a_{n})$ converges to $L$ in the earlier version if and only if it converges to $L$ in the new version; hence it is convergent, or a null sequence, in the earlier version if and only if it is in the new one.

(iii) Limits. For every convergent sequence $(a_{n})$ in $\mathbb{R}$, its limit in the earlier version is its limit in the new version.

(iv) Cauchy sequences. A sequence $(a_{n})$ in $\mathbb{R}$ is a Cauchy sequence in the earlier version if and only if it is a Cauchy sequence in the new version.

(v) Series. For every sequence $(a_{k})$ in $\mathbb{R}$, the earlier and the new version give the same partial sums (Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (i)); the series $\sum_{k=1}^{\infty}a_{k}$ converges in the earlier version if and only if it converges in the new version, and then its sum is the same in both; and it converges absolutely in the earlier version if and only if it converges absolutely in the new version.

(vi) Iterated operations. For every binary operation $\ast$ on a set $X$, every $n\in\mathbb{N}$, with $[n]$ as in Intervals of Natural Numbers (ii), and every map $a$ from $[n]$, or from a set containing $[n]$, to $X$, the iterated operation $\mathop{\ast}\limits_{k=1}^{n}a_{k}$ in the earlier version (Iterated Operations: Finite Sums and Finite Products (ii)) is the iterated operation in the new version (Iterated Operations: Finite Sums and Finite Products (ii)), and the finite sums and products of the earlier version are the finite sums and products of the new one.