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Uniform Convergence of $n^{2}(x)^{3}e^{-nx^{2}}$ on $[0,1]$

My attempt:

criterion: suppose $f_n:I\to\ J$ is a sequence of functions which converges point wise to a function $f$, then the convergence is uniform if and only if $$\lim_{n\to\infty}\sup_{x\in I}|f_n(x)-f(x)|=0$$.

I found out the point at which this function attains maximum. This happens at $x=\sqrt3/\sqrt(2n)$. and the value of $f_{n}(x)$ at this point is $(3/2)^{3/2}e^{-3/2}/\sqrt n$. The pointwise limit is $f=0$, we have $$\lim_{n\to\infty}\sup_{x\in I}|f_n(x)-f(x)|=0$$.

So, $f_{n}$ converges uniformly.

Is there a flaw in the proof? Please clarify. Also suggest any alternative proof which is more analytic rather than calculation based, I mean if bounds can be established so that I don't have to find the maxima in the interval.

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    $\begingroup$ Your approach is correct. $\endgroup$ – science Feb 13 '15 at 0:11
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Hint: The sequence of functions is eventually monotone decreasing for all $x \in [0,1]$. Then simply apply Dini's theorem.

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