Let a1,a2,...,an, A1,....,An and epsilon > 0 be real numbers such that
|ai - Ai| < epsilon for each i=1,...,n.
Then | max{a1,,,,an} - max{A1,...,An} | < epsilon.
Proof by contradiction works fine here, but can we get a direct proof ?
It looks to me like the conclusion is "< epsilon", not "< 2 epsilon". Proof:
assume wlog the max of the ai's is a1. If the max of the Ai's is A1, done.
If not, assume A2 is the max of the Ai's. Now A2 = a1 - épsilon
otherwise A2 would be smaller than A1. QED. Is that a *direct* proof?
Thank you Milton. This is OK, but isn't this also a proof by contradiction ? With a direct proof I mean obtaining an upper bound via some known inequality (e.g. triangle inequality or Schwarz-Cauchy), if it is possible of course.
Well, the first step in the proof is an excluded third "p or not p" argument, which is not
what people usually call "proof by contradiction". The other two steps are indeed formulated as contradictions, but they could easily be done by adding inequalities instead. It just looks a little neater in words.
Konrad,
you kow that for each norm N in a vector space X you have
| N(a) - N(b) |
Tank you Ulrich for seeing it in a more general context. But I guess you will agree with me that Omran's solution is very simple and elegant as it does not use any knowledge other than basic axioms of real numbers, and furthermore it is not a proof by contradiction.
One more time, Omran's proof:
Let M=max Ai, m = max ai.
Now,
1) ai < Ai+epsilon and Ai
@Konrad
In your first answer you write
and I show you that this i s possible with just to the triangle inequality you mentioned, and you only see 'a more general context'. It is exactly the context you asked for!
No question that every proof you did successfully promotes your understanding of mathematics. This applies also to Omrans 'stand alone proof'. On the other hand, mathematics is a large field and asks for an economic organization of its contents. Therefore, to recognize that some special result is a direct consequence of some major theorem that you have to know anyways is a skill that deserves cultivation.
Good point Ulrich. I didn't mean in any way to disrespect how mathematics should be done. The only reason why I do not use such way of reasoning is that I then have to introduce all these concepts of a vector space, norm, etc. just for proving this one elementary fact about real numbers. It is one small step in a proof I'm doing. I'm trying to prove it assuming as little as possible.
- Badges
- Science topic
- Similar topics
- Linguistics
- Composition Studies
- Publication