From Williams' Probability with Martingales (don't mind the red boxes).
Which part doesn't hold if $|X_k(\omega)| \le K \forall k$ but not $\forall \omega$? I guess $K$ and maybe $c$ are random, but so what? We'll just let $c$ and $K$ be positive a.s.
Consider an independent sequence $\left(X_k\right)_{k\geqslant 1}$ such that $$ \Pr\left(X_k=k^2\right)=\frac1{2k^2}=\Pr\left(X_k=-k^2\right); \Pr\left(X_k=0\right)=\frac1{k^2}. $$ Since $\sum_{k\geqslant 0}\Pr\left(X_k\neq 0\right)<+\infty$, the Borel-Cantelli lemma ensures the almost sure convergence of $\sum_{k=1}^{+\infty}X_k$. But $\sigma_k^2=1$ for all $k\geqslant 1$ hence the series $\sum_{k=1}^{+\infty}\sigma_k^2$ cannot converge.