Let $\{\ln{a_n}\}$ be a sequence of positive terms that converges to $a$. Does it imply that $\{a_n\}$ converge to $e^a$. $\epsilon$ proof?

Question

Let $\{\ln{a_n}\}$ be a sequence of positive terms that converges to $a$. Does it imply that $\{a_n\}$ converge to $e^a$. Is there a $\epsilon$ proof (formal proof)?

Answer 1

Yes, this is true. It follows from the continuity of the function $f(x) = e^x$. Whenever you have a continuous function, $\lim_{n\to \infty} f(b_n) = f(\lim_{n\to \infty} b_n)$. Hence, taking $b_n = \ln(a_n)$ we have $$\lim_{n\to\infty} e^{\ln(a_n)} = e^{\lim_{n\to\infty}\ln(a_n)} = e^a.$$

If you want an $\epsilon-\delta$ argument, then I would examine a proof concerning passing the limits through continuous functions. $e^x$ can be shown to be a continuous function via its power series definition.