I have a two part question.

1. How do I write the theorem using quantifiers?

2. How is contradiction being used with the quantifiers? Can you show with the quantified statement how we are doing proof by contradiction?

Here is the theorem from Steven Abbot’s Understanding Analysis.

Here is the theorem from Steven Abbot's Understanding Analysis.

Theorem. Two real numbers $a$ and $b$ are equal if and only if for every real number $\epsilon > 0$ it follows that $|a - b| < \epsilon$.

I have a two part question.

1. How do I write the theorem using quantifiers?

2. How is contradiction being used with the quantifiers? Can you show with the quantified statement how we are doing proof by contradiction?

Here's an image of the page: