I have $P = \{ a = a_0 < a_1 < \dots < a_n = b \}$ to be a partition of $[a, b]$. Define $\lVert P \rVert = \max \{ (a_i - a_{i - 1}): i = 1, \dots, n \} $. I need to find a sequence of partition $\langle P_k \rangle_{k = 1}^\infty$ of $[a, b]$ that satisfies $P = P_1$, $P_k < P_{k + 1}$ for every positive integer $k$ (meaning $P_k \subset P_{k+1}$) and $\displaystyle{\lim_{k \rightarrow \infty}} \lVert P_k \rVert = 0$.
The complication I have is many notations and definitions going on that I don't know where to start. I am having a toy example, say we have $[1, 2]$. Then, maybe $P_1 = \{ 1, 1.5, 2 \}$ and $P_2 = \{ 1, 1.25, 1.5, 1.75, 2 \}$, but I don't know how to evaluate the third condition, that is $\displaystyle{\lim_{k \rightarrow \infty}} \lVert P_k \rVert = 0$.
Please help me if you can. Thank you so much.
Hint: (dyadic partition)
Take $P_k = (a_0^{(k)}, a_1^{(k)}, \ldots, a_{2^k}^{(k)})$ where $a_j^{(k)} = a + \frac{j}{2^k}(b-a)$. Clearly, $\|P_k\| = (b-a)2^{-k} \to 0$ as $k \to \infty$.
Try to prove that $P_{k+1} \supset P_k$ yourself.