Hi all!

I'm here again with another Royden question. The problem's pretty simple to state:

Given a sequence of real valued functions: $\displaystyle <f_n>$ where each $\displaystyle f_n$ is continuous and defined for all real numbers, show that

the set $\displaystyle \{x:\ <f_n (x)>\ converges\}$ is an $\displaystyle f_\sigma_\delta$

I'm fairly certain that I've shown that the set is a $\displaystyle G_\delta$. And I'm pretty sure that I can prove that every G-delta is an F-sigma-delta. This would mean I'm done. But wait! Wouldn't Royden have just asked me to prove it was a G-delta in the first place? By that logic, I think I did something wrong in my proof. Casn anyone lend a hand on this problem. (Or does asnyone think I'm right about the set being a g-delta?) Thanks for any responses. They will be diligently read.

Jake