I got stuck on two problems that involves finding the pointwisse limits of piecewise functions. Hope someone can give me a hand.

1) Let

be a sequence that contains each rational in [0,1] precisely once. For each

, define

if x is irrational

if

if

Prove that

converges pointwise on [0,1] to a function that is not R-integrable.

My attempt:

I guess that this sequence of functions converges to the rational characteristic function, but I can't really show

as n approaches infinity.

I tried this again and I got this far:

if x is irrational and

if x is rational.

Consider: absolute value of

=

if x is irrational. Choose

If x is rational,

. I think f(x)=1 because eventually

as n approaches infinity.