Limit of Sequence is Limit of Real Function

Theorem
Let $\langle{a_n}\rangle$ be a real sequence.

Let $f: x \mapsto f\left({x}\right)$ be a real function.

Suppose the limit:


 * $\displaystyle \lim_{x \to +\infty} \ f\left({x}\right)$

exists.

If for every $n$ in the domain of $\langle{a_n}\rangle$:


 * $f\left({n}\right) = a_n$

then:


 * $\displaystyle \lim_{n \to +\infty} \ a_n = \displaystyle \lim_{x \to +\infty} \ f\left({x}\right)$

Proof
This is an instance of Limit of Function by Convergent Sequences, as the reals form a metric space.