Tricky problem. Any tips? Thanks SOO much!!(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Let f be continuous on [a,b] and let c be a real number. If for every x in [a,b] f(x) is NOT c, then either f(x) > c for all x in [a,b] OR f(x) < c for all x in [a,b]. Prove this using a) Bolzano-Wierstrass and b) Heine-Borel property.

2. Relevant equations

B-W property: A set of reals is closed and bounded if and only if every sequence of points chosen fro E has a subsequence that converges to a point in E.

H-B property: A subset of the reals has the HB property if and only if A is both closed and bounded.

(Note the function given fits HB and BW propertis by definition).

3. The attempt at a solution

Some hints:

For WB: suppose false. Explain how there exist sequences {x_n} and {y_n} such that f(x_n) > c, f(y_n) < c and |x_n - y_n| < 1/n

For HB: Suppose false and xplain why there should exist at each point x in [a,b] an open interval I_x centered so that either f(t)>c for all t in intersection of I_x and [a,b] or else f(t)<c for all t in the intersection of I_x in [a,b]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# IVP proof (analysis)

**Physics Forums | Science Articles, Homework Help, Discussion**