Convergence of total quadratic variation to predictable quadratic variation for continuous martingales: proof clarification.

Question

Suppose that $X\in\mathcal{M}^2_c$ is a continuous square-integrable martingale. By the Doob-Meyer decomposition there exists an increasing predictable process $\left<X\right>_t$ such that $X_t^2-\left<X\right>_t$ is a martingale. Besides the fact that $X$ is continuous implies that $\left<X\right>$ is also continuous. For $X\in\mathcal{M}^2_c$ many authors deliver the popular result $$ p-\lim_{n\rightarrow\infty}\left(X_{t_{k}}-X_{t_{k-1}}\right)^2 = \left<X\right>_t $$ with $0=t_0<t_1<\dots<t_n=t$ a partition of $[0,t]$. In proving this, the approach is the following. First it is assumed that, almost surely, $|X|\leq K$. Then a localization argument is used for the generic case. My problem is: if $X$ is a.s. continuous on the compact set $[0,t]$ shouldnt be automatic that, a.s., $|X|\leq K$ ?