It is already written using quantifiers "for all" and "there exists" but if you want $$a=b\iff(\forall\varepsilon>0:\,|a-b|<\varepsilon)$$ The direction $$a=b\implies(\forall\varepsilon>0:\,|a-b|<\varepsilon)$$ is called trivial and the direction $$(\forall\varepsilon>0:\,|a-b|<\varepsilon)\implies a=b$$ is done by proving that the negation $$(\forall\varepsilon>0:\,|a-b|<\varepsilon)\ \mathrm{and}\ a\neq b$$ leads to a contradiction.

Hope this helps. :)

It is already written using quantifiers "for all" and "there exists" but if you want $$\forall a,b:\,\Big(a=b\iff(\forall\varepsilon>0:\,|a-b|<\varepsilon)\Big)$$ The direction $$\forall a,b:\,\Big(a=b\implies(\forall\varepsilon>0:\,|a-b|<\varepsilon)\Big)$$ is called trivial and the direction $$\forall a,b:\,\Big((\forall\varepsilon>0:\,|a-b|<\varepsilon)\implies a=b\Big)$$ is done by proving that the negation $$\exists a,b:\,\Big((\forall\varepsilon>0:\,|a-b|<\varepsilon)\ \mathrm{and}\ a\neq b\Big)$$ leads to a contradiction.

Hope this helps. :)