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Real analysis- Riemann Integration.

  1. Feb 8, 2014 #1
    1. The problem statement, all variables and given/known data

    Let ##P## be a tagged partition of ##[0,3]##.
    Show that the union ##U_1## of all the sub intervals in ##P## with tags in ##[0,1]## satisfies ##[0,1-||P||]\subseteq U_1\subseteq [0,1+||P||]##. (||P|| is the norm of partition P).

    2. Relevant equations



    3. The attempt at a solution
    We first show that ##[0,1-||P||]\subseteq U_1## with tags in ##[0,1]##. Suppose that ##r\in[0,1-||P||]##. Suppose further ##||P||<1##. It follows that there exists an interval ##I_k =[x_{k-1},x_k]\in P## with tag ##t_k\leq 1## that ##r## belongs to. Since ##||P||<1## then ##x_k-x_{k-1}<1\implies x_k<1+x_{k-1}##. Since ##r\in[0,1-||P||]\implies x_{k-1}\leq r\leq 1-||P||\leq 1\implies x_k<1+x_{k-1}\leq 1+1=2\implies x_k<2## Hence ##r## is in the interval ##I_k## with tag ##t_k\in I_k##. Thus ##r\in U_1##. I think this is wrong and I'm not seeing something here. Any help would be great thanks.
     
    Last edited: Feb 8, 2014
  2. jcsd
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