I've, once again, already worked out this problem, and am just looking for somebody to validate my proof, or point out any errors. Heres the problem:
If is continuous on and , then prove there is some natural number number such that:
for some
Now, heres my attempt at proving it:
Let
We know that is continuous on . We also know that there exists a number such that:
for all
Suppose we let ; then we know that we can choose an large enough to ensure that (since for all x):
Which means that:
Which implies that:
Now, we can choose a simmilarly. Suppose , then we can ensure (by choosing a large enough or small enough ) ensure that:
Equivilantly...
Which implies that:
Now, by that one theorem (I don't remeber its name), since:
and is continuous on , and , then we a garunteed there exists some such that:
This implies that we are also garunteed an such that:
This completes the proof,.... I hope
I'm not certain if this proof is correct... I'm alittle unsure about weather or not I can garuntee some of the inequalities involving . I'm not even sure if this proof "goes in the right direction" persay. Any guidence, validation, error-pointing-out, and general advice would be much appreciated. Thanks in advance