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Math Help - Continuous Function Problem.

  1. #1
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    Continuous Function Problem.

    If f & g are continuous on R, let

    S=\{x \in R: f(x) \geq g(x) \}. Also let (s_n) \subseteq S. Suppose lim (s_n)=s, show that  s \in S.


    Well since f and g are continuous, we know

    \lim(s_n) = s then
    \lim(g(s_n)) = g(s) and \lim(f(s_n)) = f(s)

    And g(s) and f(s) exists. I just don't know how to show f(s) \geq g(s).

    Would someone please help?
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  2. #2
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    Quote Originally Posted by hockey777 View Post
    If f & g are continuous on R, let

    S=\{x \in R: f(x) \geq g(x) \}. Also let s \subseteq S. Suppose lim (s_n)=s, show that  s \in S.
    Look at that notation! There is sonething wrong with the symbols.
    Please correct it.
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  3. #3
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    Corrected.

    I guess I should also add I know the \lim(f(s_n)) \geq \lim(g(s_n))

    I'm struggling with what I should use to prove that fact.
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  4. #4
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    Suppose that g(s) > f(s) then let \varepsilon  = \frac{{g(s) - f(s)}}{2} > 0.
    \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {g(s_n ) - g(s)} \right| < \varepsilon \,\& \,\left| {f(s_n ) - f(s)} \right| < \varepsilon } \right]
    Now you also know that g(s_n ) \leqslant f(s_n ).
    From all of that a contradiction must follow.
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