I'm trying to tackle the following problem from Chung's "A Course in Probability Theory":
Consider a sequence of random variables $(X_n, n\geq 1)$, and let $S_n = \sum_{i=1}^n X_i$ denote the sum of the first $n$ random variables. Then, if $\mathbb E[X_n^2]\to 0$, we have:
\begin{equation} \frac{S_n - \mathbb E[S_n]}{n} \stackrel{\mathbb{P}}\to 0 \end{equation}
That is, we have convergence in probability.
This claim has been given a short proof here, but I don't see why the argument checks out, since it apparently uses the fact that the $1$ norm is smaller than the $2$ norm, and I don't see why that would be the case.
So far, my approach was to try to center the sequence of random variables (considering $X_n - \mathbb E[X_n]$) and try to prove $L_2$ convergence, or try to show $L_1$ convergence directly, but I was not able to do so.