What is meant by the limit of a Cauchy Sequence?

Question

I understand what a limit is, and I understand what a Cauchy Sequence is, but what is the limit of a Cauchy Sequence?

Answer 1

Suppose we have a sequence $(a_{n})_{n\geq 1}$ of real numbers. Then, the limit of the sequence is $a=\lim_{n\to\infty}a_{n}$, provided that this limit exists. In particular, if $(a_{n})_{n\geq 1}$ is a Cauchy sequence of real numbers, then it will always converge (i.e., have a limit), since all Cauchy sequences in $\mathbb{R}$ converge.

Answer 2

The "limit" of a Cauchy Sequence is the same as the "limit" of any sequence. The number "L" is the limit of sequence "$a_0, a_1, a_2, \ldots,a_n, \ldots$" if and only if, for any $\epsilon> 0$ there exist $N$ such that if $n> N$ then $|a_n- L|< \epsilon$. That is, you can get as close to $L$ as you like by taking $n$ large enough.