I was going through Durrett's probability when I was stuck in this problem. If $X_1 \ldots X_n$ are independent RVs, $S_n=\sum_{i=1}^n X_i$, $M\in \mathbb{Z}$ and $\epsilon>0$, there is this inequality:
$P(\sup_{m,n\in M} |S_m-S_n|> 2\epsilon) \le P(\sup_{m\ge M} |S_m-S_M| > \epsilon)$
I have been toying with triangle inequality to establish this, but unfortunately I cannot see this. Could someone help me please? Looks simple at first sight.