What is the definition of supremum for a sequence of real numbers?

[AxiomOfChoice]

If you're given a sequence \{x_n\}, do you have

<br /> \sup_n x_n = \lim_{n\to \infty} \left( \max\limits_{1 \leq k \leq n} x_k \right)<br />

I've never seen this definition before, but it makes sense.

...and if it's NOT the same as the supremum...what *is* it?

 

[adriank]

It is the same as the supremum. But you should try to prove they're equal for yourself; it follows somewhat easily from the definitions, where sup is the least upper bound.

 

[HallsofIvy]

Notice that this gives the supremum for a sequence- a countably infinite set. The supremum is defined for any set (assuming +\infty is a valid supremum), countable or not.

 

[Landau]

@HallsofIvy: I don't really see the relevance of your remark, but if you want to generalize: the supremum is defined for any partially ordered set.

 

[AxiomOfChoice]

@HallsofIvy: I don't really see the relevance of your remark, but if you want to generalize: the supremum is defined for any partially ordered set.

I think the relevance of his remark is this: It's not possible for the condition I listed to be the definition of the supremum of a set of real numbers because it doesn't even make sense for an uncountably infinite set; e.g., [0,1]. But I think it works just fine for any countably infinite set.

 

[Landau]

Sure it is not valid for subsets of R, in the same way it is not valid for arbitrary posets. But you were explicitly talking about sequences of reals, so...