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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: $f(\bigcup_{i} A_i)=\bigcup_{i}f(A_i)$.

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    Thank you! And for editing as well!2012-10-04

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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.

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    $A$ is an $F_\sigma$-set in $[a,b]$. Hence it's the union of countable many **closed** $K_i \subseteq [a,b]$. As $[a,b]$ is compact, each of it's closed subsets is also. So $A$ is a countable union of **compact** sets.2012-10-04