We consider a martingale $(S_n)$ with $\mathbb E(S_n^2)<K<\infty$. Suppose that $\mathrm{ Var}(S_n)\rightarrow0$.
Prove that $S=\lim_{n\rightarrow \infty}S_n$ exists and is constant a.s.
I don't know how to start here.
One thing we know is that $\mathrm{ Var} (S_n)=\mathbb E(S_n^2)-\mathbb E(S_n)^2$, so for every $\epsilon<0$, we have $\mathbb E(S_n^2)\leq\mathbb E(S_n)^2+\epsilon$. But where should I use the martingale property?
The existence of the limit $S$ follows from the martingale convergence theorem (because the expectation of $|S_n|$ is uniformly bounded), see for instance Durrett's book Probability: Theory and Examples page 235.
It remains to show that the limit is almost surely constant. To this aim, notice that $\mathbb E(S_n)\to \mathbb E(S)$ and apply Fatou's lemma to the sequence $\left((S_n-\mathbb E(S_n))^2\right)_{ n\geqslant 1} $.