My question involves supremums and their implications:(adsbygoogle = window.adsbygoogle || []).push({});

say I have the sequences [itex]\left\{x_{k}\right\}_{k=1}^{\infty}[/itex] and [itex]\left\{y_{k}\right\}_{k=1}^{\infty}[/itex]

and I know [itex]sup \left\{x_{k}:k\in N \right\} \leq sup \left\{y_{k}:k\in N \right\} [/itex]

What can I say about the sequence [itex]\left\{x_{k}\right\}_{k=1}^{\infty}[/itex]?

Can anyone suggest a text or website that goes into detail about supremums and how to use them? All my Analysis books just announce the definition of a supremum and move on, but I'd like to see a little more exposition, especially with what can be done with two sets correlated by their sups or infs as above.

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# Set of sequences, supremum

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