Let $[a,b]$ be an interval in $\mathbb{R}$. Let $P$ be a finite partition of $[a,b]$.
Define $T_n=\{a+\frac{b-a}{n}i \in \mathbb{R}| 0≦i≦n\}$ for each $n\in\mathbb{Z}^+$. (Let's denote $T_n=\{t_0,...,t_n\}$ for convenience)
My questions is, does there exists a finite refinement $Q=\{x_0,...,x_m\}$ of $P$ and $T_N$ satisfy below two properties?
That is;
(i) $i≠j\bigwedge t_i\in [x_k,x_{k+1}]\bigwedge t_j\in [x_l,x_{l+1}] \Rightarrow k≠l$
(ii) $\forall i≦N, \exists k
(iii) $\forall k
I believe this is false, but i cannot figure out how to show that..