Check for uniform convergence in the following example.

Question

Im wondering if I solved this correctly:

Clearly, $f_n(0)= n^2$ which's unbounded, so $f_n$ doesn't converge pointwise on the whole domain $R$. And since it doesn't converge pointwise, it doesn't converge uniformly. Right?

Answer 1

Indeed, we do not have pointwise convergence due to the fact that $f_n(0)=n^2$. However, when we deal with convergence, we always have to specify on which set. For example, we can show that for each positive $a$, the sequence $\left(f_n\right)_{n\geqslant 1}$ converges uniformly to $f\equiv 0$ on $[a,+\infty)$. But the convergence is not uniform on $(0,\infty)$, since the only potential limit is the null function and $\sup_{x\in (0,\infty)}\left|f_n(x)\right|\geqslant n^2$.