Prove if $X\subset \Bbb{R}$ is $F_{\sigma}$ (can be written as a countable union of closed sets) and $f$ is continuous then $f(X)$ is $F_{\sigma}$.
Proof: Let $X=\cup C_i$ where $C_i$ is closed. Then define $D_{i,n}=C_i \cap [-n,n]$. Then $D_{i,n}$ is compact (closed because intersection of two closed sets, and bounded by $n$). Note that $X=\cup D_{i,n}$. Because $f$ is continuous, it maps compact sets to compact sets. So $f(X)=\cup f( D_{i,n})$ which is a countable union of closed sets.
Does this look okay? This is a previous qualifying exam question.