Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

IVP proof (analysis)

  1. Dec 17, 2009 #1
    Tricky problem. Any tips? Thanks SOO much!! :biggrin:

    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]
  2. jcsd
  3. Dec 17, 2009 #2
    This seems like a strange question. The Heine-Borel and Bolzano-Weierstrass theorems each state that a closed and bounded subset of [tex]\mathbb{R}[/tex] is compact (each using a different characterization of compactness). However, the intermediate value theorem for closed intervals in [tex]\mathbb{R}[/tex] is not a consequence of compactness, but of connectedness.

    In your question, one might replace [tex][a,b][/tex] by [tex][0,1] \cup [2,3][/tex], a compact but disconnected set. This set possesses the Heine-Borel and Bolzano-Weierstrass properties, but the intermediate value theorem is obviously false for it (consider any function which takes one constant value on [tex][0,1][/tex] and another on [tex][2,3][/tex]).

    While the fact you are asked to prove is true, the theorems you are asked to use to prove it seem totally inappropriate to the task.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook