Proving a continuous function on [0, 1] has a "periodic equalty"?

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 (Thinking)

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