Pointwise, Uniform convergence on [a, ∞)

Question

$f_n(x) = \frac{n^2x}{1+n^3x^2}$

I claim the above function is continuous.

For $[a, ∞) , a>0$

I have tried and found out that it is pointwise convergent.

For uniform convergence, I found out that $| f_n(x) - f(x)| < \frac{1}{n^4a^3} < \epsilon$ so it is uniform convergent, but I heard that uniform convergence is only on a close interval?

Answer 1

1. Uniform convergence is not only (and not always) on closed intervals.

2. The interval $[a,+\infty)$ is actually closed.