Cramer's theorem requires MGF finite on $[0,\delta)$ for some $\delta>0$

Question

Let $X_n$ be i.i.d random variables with mean $0$ such that $\mathbb{E}e^{\lambda X_1}=+\infty$. Then for all $t>0$ $$\frac1n \log\mathbb{P}(S_n\geq nt)\rightarrow 0$$ where $S_n=X_1+\cdots+X_n$. I feel like this should be a straightforwards exercise but it's destroying my sanity for some reason. I tried to exploit the subadditivity of $\log\mathbb{P}(S_n\geq nt)$, but that wasn't particularly useful. I also suspect that the norm $\|X\|_{\psi_{1}}=\inf\{\lambda>0:\mathbb{E}e^{\frac{|X|}{\lambda}}\}$ and its connection to sub-exponential decay of $\mathbb{P}(X>t)$ might be useful. I would appreciate any input. Thanks in advance.