Limit of Sequence is Limit of Real Function
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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.): $\S 9.1$