An Exponential Inequality for Symmetric Random Variables

We prove the following exponential inequality: Let $n\geq 1$ and let $X_1,...,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have \[\mathbb{P}\big({X_1+...+X_n}\geq x,\, {X_1^2+...+X_n^2}\leq y\big)< \exp(-\frac{x^2}{2y})\,.\]