Bolzano's Theorem and continuity?
Ok so the question we were given was:
Suppose that f is continuous with |f(x)| 1 for all x R (R being the real numbers). Show that there exists some c R such that f(c) = c. Hint: apply Bolzano's theorem on a suitable interval.
I tried a few different things originally, but I just couldn't get it out, and handed up my homework. I then found out that the solution he gave was:
"Being the diﬀerence of two continuous functions, g(x) = f(x) - x is continuous with
g(2) = f(2) - 2 1 - 2 < 0
g(-2) = f(-2) + 2 -1 + 2 > 0
According to Bolzano’s theorem then, some c (-2, 2) exists such that g(c) = 0."
I don't really understand how this proves that c R such that f(c) = c, though? Also, why pick (-2, 2) as the interval for g? If someone could explain it to me please, that'd be very handy! :)