How to prove $L_{f}(P) \leq L_{f}(Q)$ when $Q$ and $P$ are partitions of $[a,b]$ and $Q \supseteq P$ I'm having trouble proving this idea.
Suppose that $f$ is bounded on the interval $[a, b]$. $P$ and $Q$ are partitions of $[a, b]$, and $Q \supseteq P$.
$$
L_{f}(P) \leq L_{f}(Q)
$$
I know that this makes sense, because by adding points to a partition, the subintervals get smaller which makes the minima $m_i$ larger, therefore making the lower sums bigger. But I'm not sure if this is a valid proof.

Let
$$m_k = \inf \{f(x): x_{k-1} \leqslant x \leqslant x_k\}.$$
If you add another partition point $\xi_k \in (x_{k-1}, x_k)$, then
$$m_k' = \inf \{f(x): x_{k-1} \leqslant x \leqslant \xi_k\} \geqslant m_k, \\
m_k'' = \inf \{f(x): \xi_k \leqslant x \leqslant x_k \} \geqslant m_k,$$
and
$$m_k'(\xi_k-x_{k-1}) + m_k''(x_k-\xi_k)\geqslant m_k(x_k-x_{k-1}).$$
Use this argument to show that
$$L_f(Q) \geqslant L_f(P).$$
