= Solution
Let $F$ be any <closed set>. For each $M>0$, choose $K_M$ as above. The probability decomposition
$$
\mathbb P(X_L\in F)\leq\mathbb P(X_L\in F\cap K_M)+\mathbb P(X_L\notin K_M)
$$
reduces the first term to a <compact set>, where the weak upper bound applies. For nonnegative numbers $u_L,v_L$, the inequality $u_L+v_L\leq2\max(u_L,v_L)$ shows that the upper exponential rate of their sum is no larger than the maximum of their upper exponential rates. Hence
$$
\limsup_L\frac1L\log\mathbb P(X_L\in F)
\leq\max\left\{-\inf_{F\cap K_M}I,-M\right\}
\leq\max\left\{-\inf_F I,-M\right\}.
$$
Let $M\to\infty$. This proves the full closed-set upper bound, including when $\inf_F I=+\infty$. The open-set lower bound was already assumed. Combining with part (b), \b[the weak principle upgrades to a large deviation principle with good rate function $I$]. This is <exponential tightness upgrades a weak large deviation principle>.
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