I have that $\{Y_i\}_{i\geq 0}$ independent random variables with $E(Y_i)=0$ and $Var(Y_i)=1$ and exist $0<M<\infty$ such that $Y_i \in [-M,M]$, for all $i \geq 1$. I need to limit $$ P\left(\sup_{n \geq 1} |S_n|>k\right)$$ where $S_n=Y_1+\cdots+Y_n$.
I need to use the Maximal Kolmogorov's inequality and then prove that $$ \displaystyle \sum_{k=0}^\infty P\left(\sup_{n \geq 1} |S_n|>k\right) < \infty. $$ But I dont know how to limite the probability using this inequality. Some tips are also valid.