# Upper bounds of a given set

1. Mar 17, 2012

### Max.Planck

#Upper bounds of a given set

In my analysis book it says:
"If a set has a upper bound, there are infinitely many upper bounds".
Why is this correct? If you have some finite set, won't there be just one?

2. Mar 17, 2012

### HallsofIvy

Re: #Upper bounds of a given set

If a is an upper bound, what about a+1, a+ 2, a+ 3, ...? Are you clear on what an "upper bound" is?

3. Mar 17, 2012

### Max.Planck

Re: #Upper bounds of a given set

Ah yes of course, I misread the definition.

4. Mar 17, 2012

### HallsofIvy

Re: #Upper bounds of a given set

Note, that if you are talking about sets of real numbers, if a set has any upper bounds, then it has a unique least upper bound.

5. Mar 17, 2012

### Alpha Floor

Re: #Upper bounds of a given set

And that unique least upper bound is called supreme.

If the supreme belongs to the set, then it's called a maximum.

EDIT: I need confirmation on the terms "supreme" and "maximum" since I've learnt Analysis in spanish and translations might not be accurate

6. Mar 17, 2012

### Office_Shredder

Staff Emeritus
Re: #Upper bounds of a given set

In english it's typically called the supremum

7. Mar 18, 2012

### HallsofIvy

Re: #Upper bounds of a given set

Which is actually Latin!!