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:

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

exists.

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

$\map f n = a_n$

then:

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


Proof

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

$\blacksquare$


Sources

  • 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 9.1$
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