In Stochastic Calculus for finance by Shreve, there is a theorem claim that (pg. 156, Theorem 3.44)
$$\langle B\rangle(T)=T,$$ or more precisely, $$\mathbb{P}\lbrace \omega \in \Omega:\langle B(.,\omega)\rangle(T)=T\rbrace=1.$$
But his proof only proves that the random variable $$Q_{\prod}-T=\sum_{k=0}^{n-1}[(B(t_{k+1})-B(t_{k}))^2-(t_{k+1}-t_{k})],$$ where $$\prod=\lbrace t_0,...,t_n\rbrace$$ is a partition of $[0,T]$, which has the properties $$\mathbb{E}(Q_{\prod}-T)=0$$ and $$Var(Q_{\prod}-T) \to 0$$ as the size of $\prod$ tends to zero.
I believe that the proof only proves $$Q_{\prod}-T \to 0 \mbox{ in }L^2$$ and $L^2$-convergence doesn't imply the almost sure convergence, am I right? If this proof is not complete, then what is the missing part?