Statement:
Let $f: D \rightarrow \mathbb{R}$ and let $c$ be an accumulation point of D. Then $\lim_{x \rightarrow c} f(x) = L$ iff for every sequence $(s_n)$ in $D$ that converges to $c$ with $s_n \ne c$ for all $n$, the sequnece $(f(s_n))$ converges to $L$.
Proof:
Suppose $\lim_{x \rightarrow c} f(x) = L$. Let $(s_n)$ be a sequence in $D$ that converges to $c$ with $s_n \ne c$ for all $n$. We must show that $\lim_{n\rightarrow \infty} f(s_n)=L$.
Since $s_n \rightarrow c$, there exists $N$ such that $n > N \Rightarrow |s_n – c| < \delta$ . Thus, for $n>N$ we have $0<|s_n – c| < \delta$, so that $|f(s_n) – L| < \epsilon$. Hence $\lim_{n \rightarrow \infty} f(s_n)=L$.
My question: The statement says if every sequence $(s_n)$ converge to $c$, but the proof gave one sequence $(s_n)$. Does one case imply all? Don't I have to find all converging sequences in the space of $D$ first?
Best Answer
When stating the definitions, theorems, properties etc. about (ALL)sequences, we are using "$a_n$","$x_n$","$a_k$",..."$s_n$" as a general sequence to make the statement. I am telling you here that the $s_n$ sequence in the proof you gave, is actually that "$a_n$"...("$s_n$") general sequence for which statements were made earlier when the sequences were initially defined and figured out. So if you prove something for $s_n$ it means you have actually proved something for sequences in general. And you may ask how do you know that is the exact same sequence, well you can compare it with those "$\text{general}$" sequences, and see if there is anything different about it. (There isn't anything different, that is why we were saying in the comments that there is nothing specific about it, and that it is actually a general sequence).
But basically when it is said "Let $s_n$ be a sequence", it is actually meant "Let $s_n$ be the same general sequence we made statements about before"(unless stated otherwise), so if we discover something new(make a conclusion) about that general sequence, that new statement actually extends to all sequences in the same way the definitions and properties of "original general sequences" were extending to all sequences.
Hope this clarifies