Proving the supremum and infimum (1 Viewer)

[BenHowe]

Hey,

I have a q as to how to prove whether the sup and inf exist for a set. I can kinda do the supremum (not really) but am clueless when it comes to the infimum...

Thanks in advance

 

[seanieg89]

For a set of what?

It is basically encoded in the definition of the real numbers that any bounded set of real numbers will have a supremum/infinum. The precise proof of this fact would depend on how you rigorously defined the reals. (And the nonrigorous high school treatment of the reals does not suffice for this purpose).

 

[BenHowe]

It's for 1st year uni. Trying to do this

 

[InteGrand]

It's for 1st year uni. Trying to do this View attachment 33843

What progress have you made? Part (a) will fall out pretty much as a consequence of the definition of sup (note that a is an upper bound of E (and since R has the least-upper bound property, sup E exists in R).)

For part (b), it's essentially because sup E may turn out to actually equal a (not be strictly less than it). You should be able to come up with an example that demonstrates this.

 

[BenHowe]

It's ok now, I just need to prove it using the two conditions but I didnt understand the second condition. Thanks for all your help