Showing that $BPP \subseteq EXP$?
A $\mathsf{BPP}$ machine $A$ has randomness complexity $r(n)$ which is polynomial in $n$. Hence, an $\mathsf{EXP}$ machine $B$ can loop over all $2^{r(n)}$ values of the random tape of $A$, simulating $A$ with each random input (each simulation taking polynomial time), and count how many times $A$ accepts or rejects. The overall decision of $A$, then, is whichever of the two counters contains a larger value at the end of this process.

Interestingly, the same reasoning also explains why $\mathsf{PP}$ is contained in $\mathsf{EXP}$.