Real Numbers Between Epsilons

Theorem
Let $a, b \in \R$ such that $\forall \epsilon \in \R_{>0}: a - \epsilon < b < a + \epsilon$.

Then $a = b$.

Proof
From Real Plus Epsilon:
 * $b < a + \epsilon \implies b \le a$

From Real Number Ordering is Compatible with Addition:
 * $a - \epsilon < b \implies a < b + \epsilon$

Then from Real Plus Epsilon:
 * $a < b + \epsilon \implies a \le b$

The result follows.