Show that function is uniformly continuous

Question

The function is defined as $f(x)=\frac{1}{1+|x|}$ and $f:\mathbb{R}\to\mathbb{R}$. I now inserted the function into the definition for uniformly continuous functions and got an expression like $\delta \cdot \frac{1}{(1+|x|)(1+|x_0|)}=\epsilon $.

How can i go on to estimate a value for $\epsilon$?

Answer 1

$$|f(x)-f(y)|= |\frac{|x|-|y|}{(1+|x|)(1+|y|)} | \le | |x|-|y|| \le |x-y|.$$