Limit of Sequence is Limit of Real Function

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

Let $f: x \mapsto \map f x$ be a real function.

Suppose the limit:


 * $\displaystyle \lim_{x \mathop \to +\infty} \map f x$

exists.

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


 * $\map f n = a_n$

then:


 * $\displaystyle \lim_{n \mathop \to +\infty} \ a_n = \displaystyle \lim_{x \mathop \to +\infty} \map f x$

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