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Use the standard Hausdorff state-space convention, so compact sets are closed. Let $F$ be any closed set and choose $K_M$ from exponential tightness. The weak large deviation principle applies to the compact set $F\cap K_M$, whileThe principle of the largest exponential term givesLet $M\to \infty$. This is the full closed-set upper bound; the open-set lower bound was already present.

For goodness, fix $r<\infty$ and choose $M>r$. The complement $K_M^c$ is open. Its lower bound and its exponential-tightness upper estimate implyThus ${\inf }_{K_M^c}I\ge M$ and $\{I\le r\}\subset K_M$. Lower semicontinuity makes this sublevel set closed, so it is a closed subset of a compact set and therefore compact. This proves that exponential tightness upgrades a weak large deviation principle to the full principle with the same good rate function.

Let $F$ be any closed set. For each $M>0$, choose $K_M$ as above. The probability decompositionreduces the first term to a compact set, where the weak upper bound applies. For nonnegative numbers $u_L,v_L$, the inequality $u_L+v_L\le 2\max (u_L,v_L)$ shows that the upper exponential rate of their sum is no larger than the maximum of their upper exponential rates. HenceLet $M\to \infty$. 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), the weak principle upgrades to a large deviation principle with good rate function $I$. This is exponential tightness upgrades a weak large deviation principle.