Is $ \left\{\frac{\sin\frac{x}{n}}{\frac{x}{n}}\right\}$ uniformly bounded?

Question

In order to calculate $$ \lim_{n\to\infty}\int_0^\infty n\sin\left(\frac{x}{n}\right)[x(1+x^2)]^{-1}dx, $$ I have the following question:

Let $$ f_n(x)=\frac{\sin\left(\frac{x}{n}\right)}{\frac{x}{n}},\quad x\in(0,\infty). $$ Is $\{f_n\}$ uniformly bounded?

It is not difficult the get a bound for every $f_n$ using $$ \lim_{y\to 0}\frac{\sin (y)}{y}=1. $$ How can I get a uniform bound?

More generally, how does one show that a sequence of functions is uniformly bounded?

Answer 1

Thanks to Daniel's comment, one only needs to show that $$ h(x)=\frac{\sin x}{x} $$ is bounded.