non-uniformly continuous function

We assert that the real function  $x\mapsto \sin \frac{1}{x}$  is not uniformly continuous on the open interval $(0,\mathrm{\hspace{0.17em}1})$.

For proving this, we make the antithesis that there exists a positive number $\delta$ such that

Choose

$$x_1=\frac{1}{\frac{\pi }{2}+2n\pi },x_2=\frac{1}{\frac{3\pi }{2}+2n\pi }$$

where the integer $n$ is so great that  ,  .  Then we have

However,

$$f(x_1)-f(x_2)=\mathrm{\hspace{0.33em}1}-(-1)=\mathrm{\hspace{0.33em}2}.$$

This contradictory result shows that the antithesis is wrong.