Convergence of a specific martingale

Question

Let $(S_n)_{n\geq 1}$ be a martingale s.t. $\mathbb{E}[{S_n}^2]<K<\infty$ for all $n$. Assume that $\operatorname{Var}(S_n)\to 0$ as $n\to\infty$. Now I would like to prove that $S_n$ converges almost surely and that the limit is a constant function. Any suggestions how this could be done? I can prove that $S_n$ is uniformly integrable and therefore, by martingale convergence thm, $S_n$ converges a.s., but how could I prove that the limit is a constant function?

Answer 1

Since $(S_n)$ is a martingale $ES_n=m$ is independent of $n$. Now, $E(S_n-m)^{2} \to 0$. So $S_n$ converges in $L^{2}$ to the constant $m$. Hence, the almost sure limit of $(S_n)$ is also equal to $m$.