# Real analysis- Riemann Integration.

1. Feb 8, 2014

### bonfire09

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