Are There Infinite Upper Bounds for a Given Set?

[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?

 

[HallsofIvy]

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

 

[Max.Planck]

Ah yes of course, I misread the definition.

 

[HallsofIvy]

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.

 

[Alpha Floor]

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 learned Analysis in spanish and translations might not be accurate

 

[Office_Shredder]

In english it's typically called the supremum

 

[HallsofIvy]

In english it's typically called the supremum

Which is actually Latin!