Thread: Maximal Element

Fix N, some integer, and suppose L is a nonempty set of integers such that every a in L is <N. Show that L has a maximal element.



How do I show this?

Originally Posted by dwsmith

Fix N, some integer, and suppose L is a nonempty set of integers such that every a in L is <N. Show that L has a maximal element.

I think you need to edit the OP.
The set you have given does not meet the description: "Fix N, some integer, and suppose L is a nonempty set of integers such that every a in L is <N"

That is verbatim from the book.

I am assuming the question goes like this.
If then define .
You can surely see that . HOW & WHY?
What can you say about

Originally Posted by Plato

I am assuming the question goes like this.
If then define .
You can surely see that . HOW & WHY?
What can you say about

N-1 can be greater than n or equal to it.

i am a bit confused. the set L = {n in Z: n < N} is the set of ALL integers less than N, whereas in the orginial question, L is presumed to be merely some (non-emtpy) set with the property that:

a in L--> a < N, which is NOT the same thing.

in other words, we are only talking about some subset of {n in Z: n < N}, and we don't know WHICH subset.

i would think it would be easier to assume our set L has no maximal element (that is, if a is in L, then there is b in L with a < b

note that a+1 ≤ b, necessarily), and from this assumption, show that some member in L must be larger than N,

which is a contradiction, so L must have a maximal element.