quadratic variation of Brownian motion $B(t)$. Let $\{X_n\}_{n \geq 1}$ be a sequence of random variables with $\mathbb{E}[X_n] = u$. Suppose $\lim_{n \to \infty}\mathrm{Var}[X_n] = 0$. Do we have that $X_n$ converges to constant $u$ almost surely?

What I ask actually comes from proving the quadratic variation of Brownian motion $B(t)$ is $t$. I was wondering how above argument for $X_n = \sum_i[B(t_i^n)-B(t_{i-1}^n)]^2$ implies that quadratic variation of Brownian motion $B(t)$ is $t$?

Yes, and the easiest way to prove this is Chebyshev's Inequality. 
Edit: Good point, but you can use Borel-Cantelli if the variances are summable. I think with your example we already know that there is an a.s. limit for the quadratic variation, so we're just making claims about what it is, I'm blanking on it now but that's likely the argument. 
