Is a Borel subset in an analytic subset of a Polish space still analytic?

Question

I encounter some problem like this

Assume $A$ is a Borel subset of $B$. $B$ is an analytic subset of a Polish space $C$. Is $A$ an analytic set in $C$?

while reading a book. But I don't know the answer. Would you please do me a favor by solving this problem for me? Thank you very much

Answer 1

Since $A$ is Borel in $B$, $A = X \cap B$ for some $X$ Borel in C. Hence $A$ is analytic in $C$.