Real Numbers Between Epsilons

Theorem
Let $$a, b \in \mathbb{R}$$ such that $$\forall \epsilon \in \mathbb{R}: a - \epsilon < b < a + \epsilon$$.

Then $$a = b$$.

Proof
From Real Plus Epsilon, $$b < a + \epsilon \Longrightarrow b \le a$$.

From Ordering is Compatible with Addition, $$a - \epsilon < b \Longrightarrow a < b + \epsilon$$.

Then from Real Plus Epsilon, $$a < b + \epsilon \Longrightarrow a \le b$$.

The result follows.