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Proving the supremum and infimum (1 Viewer)

BenHowe

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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

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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

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It's for 1st year uni. Trying to do this Capture2.JPG
 

InteGrand

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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

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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
 

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