Understanding the Completeness Axiom: A Discussion on Ross' Analysis

[Joppy]

I was reading through the early chapters of Ross' book on analysis in the section covering the completeness axiom. See below.

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Followed by a few examples.

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I'm confused as to why in the example (e), the set does not have a minimum.

I can understand that it does not have a maximum, but it seems there should be a maximum. What am i not understanding?

EDIT: The full pdf can be found https://issuu.com/juanjosesanchez22/docs/ross_sequenses.

 

[Ackbach]

It's one of those many situations in which the numbers can get arbitrarily close to something (in this case, zero), but can't actually get there. Ask yourself this: is there any natural number which, when plugged into the formula $n^{(-1)^n}$, gives you zero? But you can see from the progression that every other number is getting smaller and smaller, closer and closer to zero.

 

[Joppy]

It's one of those many situations in which the numbers can get arbitrarily close to something (in this case, zero), but can't actually get there. Ask yourself this: is there any natural number which, when plugged into the formula $n^{(-1)^n}$, gives you zero? But you can see from the progression that every other number is getting smaller and smaller, closer and closer to zero.

Ah how silly of me! For some reason i was (very foolishly) focusing on the index of numbers in the set... Thanks.