- #1
- 2,076
- 140
Homework Statement
Suppose that f(x) is a bounded function on [a,b]
If M = sup(f) and m = inf(f), prove that -M = inf(-f) and -m = sup(-f). If m>0, show that 1/f is bounded and has 1/m as its supremum and 1/M as its infimum.
Let f be integrable on [a,b]. Prove that -f is integrable on [a,b] and if m>0, prove that 1/f is integrable on [a,b].
Homework Equations
So M is the least upper bound for f on [a,b] and m is the greatest lower bound for f on [a,b].
The Attempt at a Solution
So there's lots of stuff involved in this question. I'll start by trying to prove the first thing :
If m = inf(f) and M = sup(f), prove that -m = sup(-f) and -M = inf(-f).
So we know that : m ≤ f ≤ M
Forgive me if I'm wrong, but this seems like a one liner? Simply multiplying by (-1) gives :
-m ≥ -f ≥ -M so that -m is the least upper bound for -f and -M is the greatest lower bound for -f.
Before I go any further I'd like to check if I'm not jumping the gun a bit there.
Last edited: