Math Is Fun Forum

mjcblackwood

Member

Registered: 2011-09-21

Posts: 2

Real Analysis Proof for Infimum and Supremum of functions

Consider a set X ≠ ∅. Let f and g be functions f : X → R and g : X → R both having
bounded ranges. Show that

inf{f(x) | x ∈ X} + inf{g(x) | x ∈ X} ≤ inf{f(x) + g(x) | x ∈ X}------------------------ (1)
------------------------------------------≤ inf{f(x) | x ∈ X} + sup{g(x) | x ∈ X}-------- (2)
------------------------------------------≤ sup{f(x) + g(x) | x ∈ X}----------------------- (3)
------------------------------------------≤ sup{f(x) | x ∈ X} + sup{g(x) | x ∈ X}------- (4)

Also, give examples to show that each inequality is strict.

Okay, so I really want to use the triangle inequality here, but I'm not really sure how to "throw in" absolute values since they are not present in the problem already.  Also, I don't want help on all of the inequalities here but would definitely appreciate help on the first two.  I think that once I get the idea that I will be able to do the rest of the problem myself.  Also, not really sure how to give examples as the problem that I am having with this is understanding the inf, sup on functions.  I would totally understand this problem if it were related to some sets without the functions.  Anyway, any help is really appreciated and thanks in advance!

Alg Num Theory

Member

Registered: 2017-11-24

Posts: 693

Website

Re: Real Analysis Proof for Infimum and Supremum of functions

That proves (1) and (4). The others are a tad trickier.

And

---

Me, or the ugly man, whatever (3,3,6)