I am trying to show that the for a refinement $P_0$ in the partition P then this inequality holds. $L(f,P) ≤ L(f,P_ 0 ) ≤ U(f,P_ 0 ) ≤ U(f,P)$. I understand that a refinement will make our estimate more accurate and thus the infimum will be larger and similarly for the supremum but I am unsure how to go about showing that it is the case.
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Since all partitions and refinements are finite, you can assume the refinement is by a single point (continue by induction). Let $P = \{x_1,..., x_n\}$ be a partition, $f(x_i^*) := \inf_{x \in [x_{i-1}, x_i]}f(x)$ and suppose $P_0$ is the refinement of $P$ by adding $x_{k-1} < z < x_k$ then we have: $$L(f,P) = \sum_{i=1}^n f(x_i^*) \Delta x_i \\ = \sum_{i=1}^{k-1} f(x_i^*) \Delta x_i + \sum_{i=k+1}^n f(x_i^*) \Delta x_i + f(x_k^*) (x_k - x_{k-1} + z - z)\\ \leq \sum_{i=1}^{k-1} f(x_i^*) \Delta x_i + \sum_{i=k+1}^n f(x_i^*) \Delta x_i + \inf_{x \in [x_{k-1}, z]}f(x) (z- x_{k-1}) + \inf_{x \in [z, x_{k}]}f(x) (x_k - z)\\ = L(f, P_0)$$
Same argument works for $U(f,P)$.