So I'm trying to show that:
$\lim_{k\rightarrow 0}\int_0^1\frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}} = \int_0^1\frac{dx}{\sqrt{(1-x^2)}}$
I guess this boils down to a solid understanding of uniform convergence.
There's also the second issue that the theorem in Rudin's PMA which discusses the exchange of a definite integral and a limit says that the the sequence of integrands must converge to the limit integrand uniformly on a closed interval, in this case that would be $[0,1]$. But of course our function is only defined on $[0,1)$, thus should I be considering the interval $[0,1-\epsilon]$?
As far as proving uniform convergence, I was looking at the the sequence:
$M_{a_n} = \sup_{x\in [0,1-\epsilon]}|\frac{1}{\sqrt{(1-x^2)(1-(a_n)^2x^2)}} - \frac{1}{\sqrt{(1-x^2)}}|$
for any sequence $a_n\rightarrow 0$.
And trying to prove that the sequence $M_{a_n}$ goes to zero, but it doesn't seem to upon pulling out the factor $\frac{1}{\sqrt{1-x^2}}$.
To sum up, this problem with its complexities is just a bit beyond my level of understanding and comfort with analysis. Could someone help me sort through it? Thanks.