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**I-Think** Prove $\displaystyle f(x)=\frac{1}{x}$ is not uniformly continuous over $\displaystyle (0,1)$

There are multiple ways to prove this, but I'm having trouble with a particular one

It is known that a $\displaystyle f:I\rightarrow{R}$ is uniformly continuous if and only if when $\displaystyle x_n,p_n\in{I}$ and $\displaystyle (x_n-p_n)\rightarrow{x}$, then $\displaystyle (f(x_n)-f(p_n))\rightarrow{0}$

I'm trying to use this result to prove the question

So if $\displaystyle (x_n-p_n)\rightarrow{0}$

$\displaystyle f(x_n)-f(p_n)=\frac{1}{x_n}-\frac{1}{p_n}=\frac{p_n-x_n}{x_np_n}

$

But this would seem to converge to 0, by assumption.

Can someone point out the flaw in my reasoning?

Please and thank you