 Solution

Let $T=\inf \{t\ge 0:X_t\notin (-a,b)\}$, and suppose the jumps of $X$ have absolute value at most $c$. A nonconstant centered finite-variance Lévy process oscillates, so $T<\infty$ almost surely. Before $T$ the process lies in $(-a,b)$, and at $T$ its bounded overshoot gives $X_T\in [-a-c,b+c]$. Thus the variables $X_{T\wedge n}^2$ are uniformly bounded.

Apply the optional sampling theorem for a supermartingale to the martingale from part (c):Bounded convergence theorem on the left and monotone convergence theorem on the right yield