= Solution
For every $\theta\geq0$, the <Markov inequality> and <independence> give the <Chernoff bound>
$$
\mathbb P(S_n\geq na)
\leq e^{-n\theta a}\mathbb E e^{\theta S_n}
=\exp\{-n(\theta a-\psi(\theta))\}.
$$
Optimize over nonnegative $\theta$. For $a\geq\mu$, part (a) identifies this supremum with $I(a)$, yielding the finite-$n$ estimate
$$
\boxed{\mathbb P(S_n\geq na)\leq e^{-nI(a)}.}
$$
If $I(a)=\infty$, use parameters with arbitrarily large $\theta a-\psi(\theta)$; the probability must then be zero. Taking the logarithmic upper limit proves the upper bound in the <Cramér theorem>:
$$
\boxed{\limsup_{n\to\infty}\frac1n\log\mathbb P(S_n\geq na)\leq-I(a).}
$$
For $a<\mu$, the nonnegative-parameter supremum is zero, giving the correct trivial tail-rate upper bound. \b[Exponential Markov bounds and the product moment-generating function supply the entire upper-bound argument.]
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