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A<b<c and, f is bounded on [a,b]

  1. Jul 8, 2013 #1
    1. The problem statement, all variables and given/known data
    a<b<c and, f is bounded on [a,b] and f is bounded on [b,c] prove that f is bounded on [a,c]

    3. The attempt at a solution
    there exist M1≥0 s.t. for all x ε [a,b] |f(x)|≤M1
    there exist M2≥0 s.t. for all x ε [b,c] |f(x)|≤M2

    for x ε [a,b] and x ε [b,c]
    Let M>0, and let M>M1 and M>M2
    therefore
    |f(x)|≤M1<M --> |f(x)|<M and |f(x)|≤M2<M --> |f(x)|<M
    ∴ there exist M>0 s.t. |f(x)|<M *
    so f is bounded on [a,c]

    is this proof correct? definition says f is bounded on [a,c] if M≥0 s.t. for all x ε [a,c] |f(x)|≤M
    but what I have proven is, f is bounded on [a,c] since M>0 s.t. for all x ε [a,c] |f(x)|<M :uhh:
     
  2. jcsd
  3. Jul 8, 2013 #2
    Uh...if ##x\in[a,b]## and ##x\in[b,c]##, then ##x=b##. I think you meant ##x\in[a,c]##. :confused:
     
  4. Jul 8, 2013 #3
    Nope, I meant some ##x\inℝ## lies on [a,b] and some ##x\inℝ## lies on [b,c]
     
  5. Jul 8, 2013 #4
    That's not a good way to put it. It's confusing. Try using ##x_1## and ##x_2##.
     
  6. Jul 8, 2013 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    What you should say is "if [itex]x\in [a, c][/itex] then either [itex]x\in [a, b][/itex] or [itex]x\in [b, c][/itex]". (You shouldn't use "x" to mean two different numbers.)

    Also where you say "Let M>0, and let M>M1 and M>M2" it looks as if you were "letting" M be three different numbers. Better would be "Let M> max(M1, M2)". Of course, since M1 and M2 are both positive, it follows that M> 0.
     
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