Let $f_n(x) = \frac{nx}{n^2x^2 + 1}$.
For each domain X, given below, how do you determine whether this sequence converges pointwise.
Then if it does is it possible find the limit function and determine whether the convergence is uniform or almost uniform.
X = [0,∞)
X = [1,∞)
For $x\in[1,\infty)$ we have
$$|f_n(x)|\le \frac1n$$
and the convergence is uniform on $[1,\infty)$.
For $x\in[0,1]$, there exists an $\epsilon>0$ ($\epsilon=1/2$) and a number $x\in[0,1]$ ( $x=1/n$)
$$|f_n(1/n)|=\frac12=\epsilon$$
and the convergence fails to be uniform.
But on any compact subset of $(0,1]$ the convergence is uniform. In particular, for all $\delta>0$, the convergence is uniform on $[\delta,1]$. So, $f_n(x)$ is almost uniformly convergent on $[0,1]$.