lower Darboux sum
A lower Darboux sum is the stingiest staircase over a partition: on each strip you make the rectangle only as tall as the function's smallest value there, so the staircase tucks entirely underneath the graph. It deliberately under-estimates the area, giving a floor you can trust never to rise above the truth.
Let f be bounded on [a, b] and P = {x_0, ..., x_n} a partition. On the k-th subinterval set m_k = inf{ f(x) : x_(k-1) <= x <= x_k }. The lower Darboux sum is L(f, P) = sum_{k=1}^{n} m_k (x_k - x_(k-1)). As with the upper sum, boundedness guarantees each m_k is a finite real number.
For the same partition, m_k <= M_k forces L(f, P) <= U(f, P): the floor never exceeds the ceiling. Moreover for any partitions P and Q, every lower sum is at most every upper sum, L(f, P) <= U(f, Q), because passing to the common refinement only raises lower sums and lowers upper sums. The supremum of all lower sums is the lower integral, and it squeezes against the upper integral from below.
For f(x) = x on [0, 1] with the uniform partition into n pieces, m_k = (k-1)/n (the inf is the left endpoint), so L(f, P) = sum ((k-1)/n)(1/n) = (n-1)/(2n), which increases to 1/2 — the same limit the upper sum reaches.
Lower and upper sums close the gap (n-1)/(2n) to (n+1)/(2n) as n grows, pinning the integral at 1/2.