Proving Inequalities Between Infimum and Supremum in Subset Relations

[MathematicalPhysicist]

i need to show that when A is a subset of B and B is a subset of R (A B are non empty sets) then: infB<=infA<=supA<=SupB

here's what i did:
if infA is in A then infA is in B, and by definition of inf, infB<=infA.
if infA isn't in A then for every e>0 we choose, infA+e is in A and so infA is in B, so infA+e>=infB, we can find e'>0 such that infA+e>=infB+e'>infB
so we have: infA-infB>=e'-e, let e=e'/2 then we have infA-infB>e'/2>0 so we have infA>infB. (is this method correct?).
obviously supA>=infA by definition.
i think that the same goes for supA and supB, with suitable changes.

 

[StatusX]

You don't need to use an e>0. Go back to the original definition of inf in terms of just \leq.

 

[MathematicalPhysicist]

how exactly to use it?
i mean if a is in A then a is in B, infA<=a so for every c<=a infA>=c
because a is in B also then a>=infB, but because infA is the greatest lower bound then we have infB<=infA, correct?

 

[StatusX]

Everything there sounds right except for that thing with c which I didn't follow.