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Proving that f is bounded on R

  1. Feb 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that f: R -> R is continuous on R and that lim (x -> [tex]\infty[/tex]+)(f(x) = 0) and lim (x -> [tex]\infty[/tex]-)(f(x)=0).
    Prove that f is bounded on R


    2. Relevant equations
    I have got the proof of when f is continuous on [a,b] then f is bounded on[a,b] but I'm unsure as to whether I can use this proof for this particular question.

    3. The attempt at a solution
    My attempt is merely the proof I speak of from above.
     
  2. jcsd
  3. Feb 24, 2009 #2
    but I can't just assume it is that specific function surely? plus i have to prove it's bounded, not unbounded?!?
     
  4. Feb 25, 2009 #3
    Using the definitions of the above limits you should be able to show that f is bounded on sets of the form [b,+[tex]\infty[/tex]) and (-[tex]\infty[/tex],a] for some a, b.

    Then use the result for compact intervals [a,b] to complete the proof that f is bounded on all of R.
     
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