I was solving a question on uniform convergence of sequences and series of functions and came across the following question:
Show that if $f_n(x)=nxe^{-nx^2}$, the sequence ${f_n}$ is not uniformly convergent in $[0,k)$ , $k>0$
Looking at the function, I could guess that something is not very right with $x=0$. This became my suspect!
I tried using the formulation $|f_n(x) - f(x)| < \epsilon$. f(x) is point convergent and the point-wise limit $f(x) = 0$.
This leads to : $nxe^{-x^2}$ $< \epsilon$
But I couldn't go anywhere with it. My book has provided a contradiction proof. I wanted to prove it directly. So, i plotted the function in desmos. The function shoots to infinite at x=0.
Attaching pic for reference
X axis denotes the value of x, I have set the value of n as 100000 to show the case for when n tends to infinite.
How can I show this result algebraically that 0 is a point of non-uniform convergence?
Thanks in advance!