$E\left[ S_{t} \right] = E\left[ \lim\limits_{n \to \infty}\sum_{i=1}^{n} \sqrt{t/n}~R_{i}(t) \right]=\lim\limits_{n \to \infty}\sum_{i=1}^{n}E\left[ \sqrt{t/n}~R_{i}(t) \right]=0$
Let $R_{1}(t),R_{2}(t),...,R_{n}(t)$ be a sequence of i.i.d random variables with mean 0 and variance 1. e.g ±1 fair coin flips. And $S_t=\sum_{i=1}^{\infty} \sqrt{t/n}~R_{i}(t)$. I am curious why we could interchange the summation and limit here. Could I apply monotone convergence theorem here? But the sequence $\sum_{i=1}^{n}\sqrt{t/n}~R_{i}$ is not monotonically increasing, and it could be negative as well.