Let $X_1,\dots,X_n$ be i.i.d. Poisson variables $\sim\text{Pois}(\lambda)$, with $\lambda<1$.
Now, if $X=X_1+\dots+X_n$ then $X\sim\text{Pois}(\lambda n)$ with $\mathbb E(X)=\lambda n<n$. I would like to show, perhaps using a Chernoff-style technique, that there is a constant $C$ such that $$ \mathbb P(X\geq n)<e^{-Cn}, $$ where $C$ depends on $\lambda$ but not on $n$.
What I've done: Not much: if $C$ is allowed to depend on $n$ then the claim should be easy to show: $$ \mathbb P(X\geq n)=\mathbb P(e^X\geq e^n)\leq\frac{\mathbb E(e^X)}{e^n} $$ by Markov's inequality, thus the claim holds if we pick some $C>\mathbb E(e^X)$. Except $\mathbb E(e^X)$ clearly depends on $n$ so $C$ also depends on $n$.
Any help on this is greatly appreciated.