Limit of Sequence is Limit of Real Function

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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.

$\blacksquare$


Sources