Question

Prove that Any two partitions have a common refinement.

Answer 1

Let $P$ and $Q$ be any two partitions of an interval $I$$\subseteq$ $\mathbb{R}$ such that $P$ is finer than $Q$ . Then, by the definition of common refinement, we have that $P$#$Q=\{K\cap J: K\in P, J\in Q\}$

Since $P$ is finer than $Q$ then for any $J\in Q$ $\exists$ $K\in P$ such that $J\subseteq K$. So, with this, it follows that $P$#$Q$ is not empty. Hence, any two partitions have a common refinement.