# Definition of supremum

1. Sep 2, 2010

### AxiomOfChoice

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

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

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

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

Last edited: Sep 2, 2010
2. Sep 2, 2010

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.

Last edited: Sep 2, 2010
3. Sep 2, 2010

### 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.

4. Sep 4, 2010

### 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.

5. Sep 4, 2010

### AxiomOfChoice

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.

6. Sep 4, 2010

### 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...