Let $\{X_i,i\geq 1\}$ be independent r.v.'s with $E[X_i]=0$. Then prove that for any $\epsilon>0$ there is \begin{equation*} P\{\max_{1\leq j\leq n}S_j>\epsilon\}\leq 2P\{S_n\geq\epsilon-E|S_n|\}. \end{equation*}
Applying Feller-Chung lemma with $A_j=\{S_j>\epsilon\},B_j=\{S_n-S_j>-E|S_n|\}$ we can obtain that \begin{equation*} P\{\max_{1\leq j\leq n}S_j>\epsilon\}\inf_{1\leq j\leq n}\{P[B_j]\}\leq P\{S_n\geq\epsilon-E|S_n|\} \end{equation*} But now the question is, how to show the term $\inf_{1\leq j\leq n}\{P[B_j]\}\geq\frac{1}{2}$ without information about higher order moment?