Sub-Gaussian Sum with a Volatility Proxy

Suppose $X_1,\dots,X_n$ are independent centered random variables and each satisfies \[ E[e^{tX_i}]\le e^{\sigma^2 t^2/2}\qquad\text{for all }t\in\mathbb R. \] Show that for $S_n=\sum_{i=1}^n X_i$, \[ P(S_n\ge x)\le \exp\!\left(-\frac{x^2}{2n\sigma^2}\right). \]