If $(x_n)$ is a sequence and $l \in \mathbb{R}$ prove $\displaystyle\lim_{y\to\infty}f(y)=l$ iff $\displaystyle\lim_{n\to\infty}x_n=l$

Question

Let $(x_n) \subset \mathbb{R}$ be a sequence of real numbers and $l \in \mathbb{R}$
Define $f:\mathbb{N} \rightarrow \mathbb{R}, n \mapsto f(n):=x_n$
Prove that $\displaystyle\lim_{y\to\infty}f(y)=l$ iff $\displaystyle\lim_{n\to\infty}x_n=l$

I have no idea where to begin; I'm not even sure why we can discuss $\displaystyle\lim_{y\to\infty}f(y)$

Answer 1

First we have,

$$\lim_{y\to \infty} f(y)=\lim_{n\to \infty} f(n)$$

since the name of variable does not change the limit definition (as you can check with $\epsilon$-$\delta$ usual definition). By definition of the function $f$ we have that $f(n)=x_n$ for all $n\in\mathbb{N}$ then $$\lim_{y\to \infty} f(y)=\lim_{n\to \infty} f(n)=\lim_{n\to \infty} x_n$$

finally $$\lim_{y\to \infty} f(y)=l\Leftrightarrow \lim_{n\to \infty} x_n=l.$$