The following is an exercise from Pollard's "A user's guide to measure theoretic probability".
I have to clarify that Pollard uses $\mathbb{P}$ to denote expectation and he omits the indicator function from sets meaning that $\{ \max_{i \leq n} |X_i| \leq \epsilon \}$ stands for $1\{ \max_{i \leq n} |X_i| \leq \epsilon \}$. Here is my attempt to tackle this following these hints. We define the stopping time
$$ \tau = \begin{cases} \inf\{ i \leq n: |X_i| > \epsilon \} \\ n \quad \text{if} \ |X_i| \leq \epsilon \ \text{for all} \ i \leq n \end{cases}. $$
Then $\{ \max_{i \leq n} |X_i| \leq \epsilon \} \subset \{ \tau = n \} $ so that
$$ V_n 1 \{ \max_{i \leq n} |X_i| \leq \epsilon \} \leq V_{\tau} 1 \{ \tau = n\} $$
and, by our assumptions,
$$ \mathbb{E} V_\tau \leq \mathbb{E} (1+\epsilon)^2 = (1+\epsilon)^2, $$
as $|X_\tau| \leq |X_{\tau-1} - X_{\tau}| + |X_{\tau-1}| \leq 1 + \epsilon $ and $\tau$ is finite. Combining these facts now yields the desired result. Is my reasoning correct? Thank you.