Limit of probability expression

Question

Let $X$ be a real random variable having $E(X^2)<+\infty$.

Let $\varphi(t)$ be the caracteristic function of $X$.

I need to show that $\lim_{t\to 0}2\frac{1-\Re(\varphi(t))}{t^2}=\mathbb{E}(X^2)$.

What I did:

We know that $\lim_{t\to 0} 2\frac{1-\cos(tx)}{t^2}=x^2$ Is that enough, with the linearity of the expectation to prove that $\lim_{t\to 0}2\frac{1-\Re(\varphi(t)}{t^2}=\mathbb{E}(X^2)$?

Thank you!

Answer 1

You have to justify taking the limit inside the integral. For that use the fact that $1-\cos x \leq \frac {x^{2}} 2$ and use Dominated Convergence Theorem.