TITLE: Proof that limits are unique
QUESTION [5 upvotes]: I have been working through a few calculus proofs, and to prove that a limit was unique i assumed the fact that if $c,d\in\mathbb{R}$ if given any $\epsilon>0$ we have $|c-d|<\epsilon$, then $c=d$.  However, is there an actual proof of this statement, or is this more of an obvious reality?

REPLY [2 votes]: This is actually not provable without really getting deep into the definition/construction of $\mathbb R$. We need the fact that if $a \neq b$, then $|a-b|>0$, and proving that, rather than taking it as an axiom, requires rather complicated mathematics.