Proving infimum

[mdrak12]

Let B= x²+2x-3
I've proven that -4 is a lower bound for set B. I now need to prove -4 is the infimum of the set. I understand why -4 is the infimum, its the formally proving that it is the infimum.

These are my thoughts so far,

def. infimum
i) must be a lower bound for set B
ii) if k>lower bound, then there is some x within the set such that x <k (so k is not a lower bound)

If k > -4, want to show k is not a lower bound of B,
So do I take x²+2x-3 < k and solve for x?

OR

Assume k is a lower bound and k > -4.
Then there is some x within the set such that x < k. Define that x = (k-4)/2. Then do I go from there?

 

[Steven G]

let E represent epsilon.
Let E>0. Now show that -4-E is NOT a lower bound.
Now how can a number, -4-E or not, not be a lower bound x²+2x-3? The answer is quite simple, you find a number in x²+2x-3 that is even lower that -4-E

So how can you find a number between -4-E and -4? One way is to compute the average! . Call this number D for a moment (but actually find it). Just find the x-value that makes x²+2x-3 =D