I need help with this problem.
Give an example of a sequence of integrable functions fn:[a,b]-->R convergent (pointwise) to a non-integrable function f:[a,b]-->R.
What kind of integration? Riemann, Lebesgue, something else?
For Riemann integration, enumerate the rationals in the interval as $\displaystyle \{q_i\}_{i=1}^\infty$, then let
$\displaystyle f_n(x)=\begin{cases}1&x\in\{q_1,\dots,q_n\}\\0&\te xt{otherwise.}\end{cases}$
Then $\displaystyle \int f_n=0$ for every finite n, but the limit function is not integrable.
For any integral: choose a non-integrable nonnegative function $\displaystyle f:[a,b]\to\mathbb{R}_+$ (any one you want), then $\displaystyle f_n=\min(f,n)$ is bounded by $\displaystyle n$, hence integrable on $\displaystyle [a,b]$, and converges pointwise to $\displaystyle f$.
I don't think that from the assumption that f be non-negative, non-integrable it follows that $\displaystyle f_n =\min(f,n)$ is integrable.
For the Riemann integral it is easy to give an example of a non-negative and bounded function that is non-integrable (e.g. the characteristic function of $\displaystyle \mathbb{Q}$), and in the case of the Lebesgue integral we must consider the possibility that f might not even be measurable.
So, I think, one would have to be a bit more specific about the cause of the non-integrability of f.