Consider a geometric scaled walk,
\begin{align*} X_k &= X_0\exp[\frac{1}{\sqrt{k}}(S_k)]\\ &= X_0\exp[\frac{1}{\sqrt{k}}(2T_k - k)] \end{align*}
where $T_k \space \tilde{} \space Binomial(k, 0.5)$
How can I use the Central Limit Theorem to show that the cdf of $\frac{S_k}{\sqrt{k}}$ converges to that of the standard normal as $k \rightarrow \infty$? Is this is the same as showing that a random walk converges to the Brownian motion in continuous time?
I can't seem to find a formal proof of this result anywhere so any guidance would be much appreciated.