Sequences and uniform convergence

Question

I have the following sequence of functions: $f_n(x) = \frac{x^2 + 4}{2x^2 + (nx + 2)^2}$

How do I show that it converges uniformly on $[1, \infty]$?

I know that I need to show that there exists some $N \in \mathbb{N}$ such that $|f_n(x) - f(x)| < \varepsilon$ for all $n > N$ but I'm not sure how to bound the sequence.

Answer 1

$0 \leq f_n(x) \leq \frac {x^{2}+4} {n^{2}x^{2}} \leq \frac 5 {n^{2}}$ since $\frac {x^{2}+4} {x^{2}}=1+\frac 4 {x^{2}} \leq 1+4 =5$. Hence $f_n \to 0$ uniformly.