For a given sequence $\left\{x_n\right\}$ we define its sequence of geometric means to be $\left\{G_n\right\}$, where $G_n=\sqrt[n]{x_1x_2 \cdots x_n}$. We have to show that if $x_n \rightarrow L$ then $G_n \rightarrow L$.
One method that I have implemented to prove this theorem is that I have used the property of continuous function (i.e if $f : A \rightarrow \mathbb{R}$ is a continuous function, then a convergent sequence $\left\{x_n\right\}$ in $A$ implies $\left\{f(x_n)\right\}$ is convergent, and $f(x_n) \rightarrow f(x)$ if $x_n \rightarrow x$ ) and the fact that the sequence of arithmetic mean of a convergent sequence preserves its limit (same statement as in the paragraph above but it's replaced with arithmetic mean in place of geometric mean, which I've proven).
But suppose you don't know anything about continuous function. Can anyone give me a hint as to how could I solve my original problem, without using any properties of continuous function ?
