During a lecture on $L^p$ spaces, the lecturer made a few comments about the sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ that I am not sure I fully understand.
1) First he said $f_n\in L^2(0,1)$:
I have $\int_0^1\frac{n^2}{1+2n\sqrt{x}+x}=n^2\int_0^1dmx+\frac{n}{2}\int_0^1\frac{1}{\sqrt{x}}dmx+n^2\int^1_0\frac{1}{x}$. Now, the very last integral does not converge, so this doesn't seem to me to belong to $L^2(0,1)$, what am I missing?
2) Then his powerpoint presentation asked whether the pointwise limit of $f_n(x)$, $f(x)=\frac{1}{\sqrt{x}}$, belongs to either $L^2(0,1)$ or $L^1(0,1)$, or both.
Am I right in saying that it belongs to $L^1(0,1)$ but not $L^2(0,1)$ since the integral would converge in the first case and not in the second?
3) Then he asked to determine whether $f_n$ converges to $f$ in $L^1(0,1)$ or $L^2(0,1)$.
Since $f$ is not in $L^2(0,1)$, the sequence $f_n$ converges to $f$ in $L^1(0,1)$. Is that all there is to this question?
4) Finally, he asked whether $f_n$ is a Cauchy sequence in either $L^2(0,1)$ or $L^1(0,1)$.
My understanding is that since all convergent sequences are Cauchy, $f_n$ is Cauchy in $L^1(0,1)$, and since all $L^p$ spaces are complete (Banach), then not converging in $L^2(0,1)$ implies that $f_n$ is not Cauchy in $L^2(0,1)$
I know this is wordy, but I want to make sure I understand this stuff, any comments or insights are appreciated.