# Unique infimum proof

1. Mar 15, 2012

### jaqueh

1. The problem statement, all variables and given/known data
Let A be a subset of ℝ, now prove if inf(A) exists, then inf(A) is unique

3. The attempt at a solution
I am using an epsilon proof but i dont think i am going about it the right way, can someone nudge me in the right direction

2. Mar 15, 2012

### Ray Vickson

How you do it depends very much on what *definition* you are using for "inf". Different (but equivalent) definitions would need different types of proofs.

RGV

3. Mar 15, 2012

### Robert1986

Usually, for something like this, it is best to start of saying "Let x and y be inf(S)" and then show that x and y are, in fact, the same. Now, showing that |x - y| < epsilon might be a good strategy, but there are probably more efficient methods. But, as RGV said, it depends on how inf is defined for you. If you have defined it as a greatest lower bound, the method I mentioned works very well.

4. Mar 15, 2012

### jaqueh

yes i think im going to revise it as x and y are infimums for A then x<y and y<x so x=y

5. Mar 15, 2012

### scurty

That's a contradiction the way you wrote it. Did you mean $x \leq y, \ \text{and} \ y \leq x,\ \text{so} \ x=y$?

6. Mar 15, 2012

### jaqueh

yes that is what i meant and i did write it that way x≥y y≥x