Lipschitz functions carry $F_\sigma$ to $F_\sigma$. Let $f:[a,b] \rightarrow \mathbb{R}$ be a Lipschitz function.  I want to show that it carries $F_\sigma$ sets to $F_\sigma$ sets.
I'm not sure how to demonstrate this.  Specifically I'm not sure what property of continuity or Lipschitz would preserve the $F_\sigma$ property.  I do know that this is true: $f(\bigcup_{i} A_i)=\bigcup_{i}f(A_i)$.

Hint: A closed set $K \subseteq [a,b]$ is compact (as $[a,b]$ is). Hence its image $f[K]$ under the continuous function $f$ is compact also.
