Suppose $X_n\sim N(0,1) $ is iid, then it is easy to see that $$\sum_{n=1}^{\infty}\frac{X_n}{n}\cos nx$$ converges a.s. for any $x$ since $$\sum_{n=1}^{\infty}var(\frac{X_n}{n}\cos nx)<\infty$$ but how to show that the convergence is uniform?

Suppose $X_n\sim N(0,1) $ is iid, then it is easy to see that $$\sum_{n=1}^{\infty}\frac{X_n}{n}\cos nx$$ converges a.s. for any $x$ since $$\sum_{n=1}^{\infty}var(\frac{X_n}{n}\cos nx)<\infty$$ but how to show that the convergence is uniform? That is, for a mesurable set $A\subset \Omega, \mathbb{P}(A)=1$, the series converges for all $x\in\mathbb{R}$ and $\omega\in A$.