Given: Suppose that $f$ is a bounded function defined on [a, b] and that $P = \{x_0, …, x_n\}$ is a partition of [a, b]. For each i = 1, …, n we let $$M_i(f) = sup \{ f (x) : x ∈ [x_{i –1}, x_i]\}$$ and $$m_i(f) = inf\{ f (x) : x ∈ [x_{i –1}, x_i]\}$$ When only one function is under consideration, we may abbreviate these to $M_i$ and $m_i$, respectively. Let $ \Delta x_i = x_i – x_{i-1}$ for (i = 1,…, n), we define the upper sum of $f$ with respect to partition $P$ to be $$U(f,P) = \sum_{i=1}^n M_i\Delta x_i$$ and the lower sum of $f$ with respect to $P$ to be $$L(f,P) = \sum_{i=1}^n m_i\Delta x_i$$
Since we are assuming that $f$ is a bounded function on [a, b], there exist numbers $m$ and $M$ such that $m ≤ f (x) ≤ M$ for all $x$ ∈ [a, b]. Thus for any partition $P$ of [a, b] we have $$m(b − a) ≤ L( f ,P) ≤ U( f ,P) ≤ M(b − a).$$
My Question: What does $m(b-a)$ (and $M(b-a)$) refer to? As far as I can tell $m$ is the minimum value of a function $f$ and $M$ the maximum value of a function $f$.
Edit: I understand that $b-a = (\Delta x_1+...+\Delta x_i)$. I just do not understand where this $m()$ and $M()$ come from.
Edit 2: The question has been answered. I misinterpreted $m(b-a)$ as a function instead of $m \cdot (b-a)$