Chebyshev's Inequality Improvement

Question

I'm having trouble improving the upper bound on Markov/Chebyshev's inequality in this particular example:

Show that $$\lim_{n\to\infty} n\mathbb{P}(|X_1|\geq \epsilon\sqrt{n})=0$$

Clearly, Markov's inequality yields the upper bound $n\mathbb{E}[|X_1|^2]/\epsilon^2$ which is not good enough. I'd really appreciate any hints or direction.

Answer 1

Note that $nP(|X_1| > \sqrt{n}) = E[n1(X_1^2 > n)]\leq E[X_1^21(X_1^2 > n)]$. The quantity inside the expectation is bounded by $X_1^2$ and converges to zero almost surely. Now use the dominated convergence theorem.