I am trying to prove the Itō's lemma, and need to show that if $X$ is a semimartingale and $f$ is a $\mathcal{C}^2$-function, then $f(X_t)$ is again a semimartingale.
How do I do that? I cannot see how to decompose $f(X)$ into a martingale and a process of bounded variation (even for $f(x) = x^2$ this gets tedious). Or is there some kind of completeness result that ensures the existence of a decomposition?