Possible Duplicate:
Proof with inequalities
I've just started reading a book on real analysis and a lot of my proofs reduce to proving this fact over and over again:
For all $\epsilon > 0$, if $a < b + \epsilon $ then $a \leq b$. How do I do this. Are there different ways of doing it?
Thanks!
The statement should probably read something like "If for all $\epsilon > 0$, $a < b + \epsilon$, then $a \le b$."
There are numerous ways to convince yourself of this. One is to examine the contrapositive: if $a > b$, take $\epsilon > 0$ so that $a > b + \epsilon$ (i.e., $\epsilon < a-b$). Then it's not true that $a < b + \epsilon$ for all $\epsilon > 0$. An easy modification of this will give you a proof by contradiction.
