You have us at distinct disadvantage: you have the book and we donít.
Moreover, as stated that statement is false.
That function has no maximum on
Do you mean continuous function?
in trying to prove that a function on a closed bounded interval attains a max value.
i was reading a proof and the way they proved it was that to show that
let M be the sup of f then show that
f(x) is less than M-1/M_1 hence M-1/M_1 is an upper bound for f, which is smaller than M..hence M is the max..
but im wondering, if that works, then M-1/M_1 should be the max of f right? and not M..
i think i get what you are saying..
but for this method that i have stated above where M is assumed to be the sup of f since f is continuous and hence bounded,
showing that f(x) is less than M-1/M_1 hence M-1/M_1 is an upper bound for f doesnt show that f attains the sup M right?
what the book did was this:
let M be the sup of f. thus f(x)<M
since f(x) is cont, then g(x)= 1/(M-f(x)) is cont
since g(x) is cont, it is bounded and let g(x)< M_1
thus max of f(x)=M
To prove that is bounded, suppose not. Then, there is a sequence with . Recall though that by the Bolzano-Weierstrass Theorem we know that has a convergent subsequence , and in particular is Cauchy. Recall though that every continuous function on a closed and bounded interval is uniformly continuous (this is the Heine-Cantor theorem) and that uniformly continuous functions preserve Cauchy sequences, and so is Cauchy. But, this is clearly impossible since . It follows that there exists no such sequence and so is bounded.
So, we know now that exists. Suppose then that ]math]f(x)<M[/tex] for all . Then, the function is continuous. Thus, by our last paragraph we know that is bounded above by . So, solving for gives for all . But, this contradicts that . So, for some .
Sorry, I missed the function in the sequences. Let me rewrite the argument correctly and fully.
We found a sequence such that for all ; the sequence diverges to infinity. Note that if we take any subsequence of this, , it too will diverge, since for all .
Now from the Bolzano-Weierstrass theorem, the sequence has a convergent subsequence, call it . Since is continuous, . In other words, we found that the subsequence converges, which contradicts our statement above.
You need to mention that is closed and thus since the real numbers are first countable that so that is defined at . Otherwise, your argument is incorrect. Then, with this I don't feel as though your argument is any shorter than mine.
You are right. I implicitly used closure here: . (I specifically would like to point out that you can prove this simply by using the fact that is a "closed" interval. First-countability or the topological concept of closure is not necessary.) This allows me to make the continuity claim, that . Other than this, I do not see why the argument is not sound.
My main issue with your proof was that it calls on some "heavy" tools: uniform continuity and Cauchy sequences. The proving their associated results actually takes some work, and therefore leads to a longer proof.
I meant no insult to your competency. I'm just trying to be very clear about where I am attempting to shorten the proof.
I think that the issue boils down to what we mean by "shorter proof". If we mean less words, then I can agree that after all of the details are written, our two approaches are more or less the same. However, when I said "shorter", I meant that it uses less tools and results; in other words, it takes less to make it self-contained. (I think that it is better of have a more elementary proof - though, this could be a matter of opinion.) Your method uses sequences, Cauchy sequences, continuity, and uniform continuity and some results. I'm sure that you would agree that this is much more than my approach, which uses sequences, continuity and some results.