For every $\epsilon\gt 0$, $|a-b|<\epsilon $ ,then b=a .

Question

I have done a proof by myself but not sure about it

proof: $|b-a|<\epsilon $ =$a-\epsilon $

Answer 1

We have $\forall \epsilon > 0,\quad |a-b| < \epsilon$

If we suppose that $a \neq b$ then $|a-b| \neq 0$, we choose $\epsilon = \dfrac{|a-b|}{2} > 0$

Then $|a-b| < \dfrac{|a-b|}{2} \implies 1 < \dfrac{1}{2}$, contradiction!

Conclusion : $\forall \epsilon > 0,\quad |a-b| < \epsilon \implies a =b$