convergence of sum of i.i.d random variables

Question

(Durrett 3.3.21) I want to show the following;

Let $X_1,X_2,...$ be i.i.d random variables. If $S_n=X_1+X_2+...+X_n$ converges in distribution, then it converges in probability.

I have a hint saying that if $m, n \rightarrow\infty,$ then $S_m-S_n\rightarrow0$ in probability. -(1)

Now use Exercise 2.5.11 in Durrett's Probability: Theory and Examples. -(2)

'Let $X_1,X_2,...$ be i.i.d random variables and $S_n=X_1+X_2+...+X_n$. If $S_n/n \rightarrow 0$ in probability, then $\max_{1\le m\le n}(S_m)/n \rightarrow 0$ in probability. (Durrett ex 2.5.11)'

I have proved hint (1), but couldn't connect it with hint (2).