$f(x)=\frac{x}{1-|x|}$ is not uniformly continuous

Question

I'm trying to prove that $f:B\to \mathbb R^n$, $f(x)=\frac{x}{1-|x|}$, where $B\subset R^n$ is an open ball centered at origin and radius $1$ is not uniformly continuous.

I've already tried find some sequences such that this propriety fail:

$\lim (x_n-y_n)=0\implies \lim(f(x_n)-f(y_n))=0$

or find some uniformly continuous function such that its composition with $f$ is not uniformly continuous.

Are there another techniques I can use to prove this function is not uniformly continuous?

Thanks in advance

Answer 1

You said you tried to find sequences where $x_n-y_n\to 0$ but $f(x_n)-f(y_n)\not\to 0$. How about these sequences? $$x_n: .9, .99, .999, \dots$$ and $$y_n: .99, .999, .9999, \dots$$