# User:IssaRice/Taking inf and sup separately

This page describes a trick that is sometimes helpful in analysis.

## Contents

## Satement

Let and be bounded subsets of the real line. Suppose that for every and we have . Then .

Actually, do and have to be bounded? I think they can even be empty!

## Proof

Let and be arbitrary. We have by hypothesis . Since is arbitrary, we have that is an upper bound of the set , so taking the superemum over we have (remember, is the *least* upper bound, whereas is just another upper bound). Since was arbitrary, we see that is a lower bound of the set . Taking the infimum over , we have , as required.

## Applications

### liminf vs limsup

(Notation from Tao's *Analysis I*.)

Let be a sequence of real numbers. Let and let . Then we have .

Consider the sequences and defined by and .

Now consider the sets and . If we can show that for arbitrary , then we can apply the trick to these sets to conclude that .

### Lower and upper Riemann integral

(Notation from Tao's *Analysis I*.)

Let be a bounded interval on the real line, and let .

We have

We want to show .

Define

Then we have and . To apply the trick all we need to do is to let be a p.c. function on that majorizes , and let be a p.c. function on that minorizes , and show that .

## alternating series test

(this one is more of a failed application)

each even partial sum is at least as large as each odd partial sum, so the inf over the even partial sums is at least as large as the sup over the odd partial sums. this actually isn't strong enough to prove what we want. we actually need the stronger condition that the even partial sums are a decreasing sequence, and that the odd partial sums are an increasing sequence, and that eventually their difference becomes arbitrarily small.

## References

After I wrote this page, I found the same theorem in Apostol's *Calculus* (volume 1, 2nd edition, p. 28) in the section "Fundamental properties of the supremum and infimum".