Proving Borel Set B: Summation of Borel Functions and Lebesgue Measure Y

[hellbike]

let f_n be series of borel functions. Explain why set B = {x: \sum_n f_n(x) is not convergent} is borel set.

Proof, that if\int_R |F_n|dY \leq 1/n^2 for every n then Y(B) = 0.Y is lebesgue measure.for first part i thought that set of A={x: convergent} is borel, and B=X\A so it's also borel, but i got 0 points, so I'm wrong.

for second part - it seems quite obvious for me that for every x \neq 0
lim_n Y(f^{-1}_{n}[x])->0 and i think proving this would be enough.
I tried doing this using simple functions, but got 0 points.

 

[ystael]

For the first part, the problem is likely that you didn't show why the set of points where (f_n) converges should be a Borel set. It's not completely obvious.

 

[hellbike]

and for the second part?