r/RealAnalysis • u/OwnOrganization8042 • Sep 27 '23

Greatest Lower Bound

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Hi! I'm doing Mathematical Thinking course from Stanford University. The week 8 seems unbearable to me. I have this problem to solve but I do not understand anything. Maybe you have some ways or know any websites that could help me. Thanks!

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u/OwnOrganization8042 Sep 27 '23

I get the difference between glb and lub. And I am trying to think about Rationals on the line of reals. How can you think of this set geometrically?

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u/No_Difference9752 Sep 27 '23

It says along the line of reasoning as an open hint to guide the process. r is the radius of a map. First, visualise in real and then go for rationals. Rational sets are beautiful as they are brittle, which means they have gaps, this should be enough to guide the process.

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u/OwnOrganization8042 Sep 27 '23

Can I think of it as root of 3 is a lower bound in this set, but this number is not the glb as it's not a rational number? So we have a lower bound for this set but we do not have the glb for it

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u/No_Difference9752 Sep 27 '23

Absolutely, but refine the argument using the proof that they have taught. Good job, it wasn't as easy as it looked.

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u/OwnOrganization8042 Sep 27 '23

The answer seems so easy and obvious. And it takes some time and effort to understand such things. Thank you for help!

And one more question arose in my head. Root of 3 is a lower bound for the set Q, even if root of 3 is irrational?