Prove computation of reciprocal unstable\limit doesn't exist

Question

Let $f=\frac{1}{u}, u \neq0, u<10^{-3}$.

Proof: Let $u_1=10^{-7}, u_2=10^{-7}+\frac{\delta}{3}$, and $\epsilon=10^{-5}$. Then, $|u_1-u_2|=10^{-7}-[10^{-7}+\frac{\delta}{3}]=|-\frac{\delta}{3}|=\frac{\delta}{3}<\delta$.

Now I need to prove that $|\frac{1}{u_1}-\frac{1}{u_2}|>\epsilon$ - and I am completely stuck.

Answer 1

Hint:

$|\dfrac{1}{u_1}-\dfrac{1}{u_2}|<\epsilon \iff |\dfrac{u_2-u_1}{u_1u_2}|<\epsilon \iff |u_2-u_1|<\epsilon|u_1u_2| $