Hello chaps
The question is as follows:
My half-baked idea:Use the well ordering principle to prove the maximum principle
I'm not sure how we can go further with this. What else can we prove with this information? What am I missing?Let:
and
(all natural numbers)
There is a hint which states we should let a new setbe the upperbounds of
in
(natural numbers)
We know, so by WOP it must have a minimum, which we shall call
.
We then assume:
That was the exact question in my practical. My lecturer is German though. Perhaps his English is not entirely correct? I wouldn't know.Originally Posted by Plato
It's actually quite simple though. Prove that if you have a subset (of the natural numbers) which is non-empty and bounded, that the subset must have a maximum.
Then basically you assume it does not have a maximum and do a proof by contradiction that the earlier assumption was incorrect.
To continue from yesterday:
then clearly:
Which meansmust be a maximum for the set
IfHere would be my approach.
Because, then by W.O.
has a least (or first) term. Call it
Ifwhat does that mean?
Ifis
In either case doesthen surely
Ifthen I don't think
(not impossible, but not definitely) unless we have some indication that there is some
where
WellIfwhich means
.
Sois not an upper bound of
So some member ofhas the property that
.
But there is no integer between. Thus ??
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