Determine whether the functional sequence is uniformly convergent.

Question

The functional sequence is given by; $$f_n(x)=ne^{-(x-n)^2}$$ So far all I have got is $$f(x)= \lim_{n\to \infty} ne^{-(x-n)^2} = 0 $$ i.e. it is pointwise convergent to zero. Question: Am I correct here and do I then use $ \vert f_n(x)-f(x)\vert\lt\epsilon$ to determine whether the convergence is uniform?

Answer 1

$$ f_n\left(n\right)=n $$ Hence

$$\underset{x \in \mathbb{R}}{\text{sup}}\left|f_n\left(x\right)-f\left(x\right)\right| \geq n $$ So it does not tend to $0$. There's no uniform convergence.