Let f, g:[a, b] →ℝ be bounded functions. Show that L(f)+L(g) ≤L(f+g) and U(f+g) ≤U(f)+U(g)

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Let f, g:[a, b] →ℝ be bounded functions. Show that L(f)+L(g)...

Let $f, g:[a, b] \rightarrow \mathbb{R}$ be bounded functions. Show that $$ L(f)+L(g) \leq L(f+g) \quad \text { and } \quad U(f+g) \leq U(f)+U(g) $$ Hence conclude that if $f$ and $g$ are integrable, then so is $f+g$, and the Riemann integral of $f+g$ is equal to the sum of the Riemann integrals of $f$ and $g$.

Let $f, g:[a, b] \rightarrow \mathbb{R}$ be bounded functions. Show that
$$
L(f)+L(g) \leq L(f+g) \quad \text { and } \quad U(f+g) \leq U(f)+U(g)
$$
Hence conclude that if $f$ and $g$ are integrable, then so is $f+g$, and the Riemann integral of $f+g$ is equal to the sum of the Riemann integrals of $f$ and $g$.

Key Concepts

Riemann Integrability

Riemann integrability is a property of a bounded function on a closed interval [a, b] that allows it to be approximated arbitrarily well by sums over partitions. This is achieved by considering the upper and lower sums of the function, and a function is Riemann integrable if the difference between the upper and lower sums can be made as small as desired by choosing an appropriate partition.

Lower Darboux Sum

The lower Darboux sum of a function over a partition is the sum obtained by taking, for each subinterval of the partition, the product of the subinterval's length with the infimum of the function over that subinterval. It provides a lower bound on the area under the curve, and the supremum of all lower sums over all partitions is used to define the lower integral.

Upper Darboux Sum

The upper Darboux sum of a function over a partition is calculated as the sum of the products of the lengths of the subintervals with the supremum of the function on those subintervals. It gives an upper bound on the area under the curve, and the infimum of all upper sums over all partitions is used to define the upper integral.

Linearity of the Riemann Integral

The linearity property of the Riemann integral states that the integral of a sum of functions is equal to the sum of their integrals, provided that the functions involved are integrable. The inequalities involving lower and upper sums, such as L(f) + L(g) ? L(f+g) and U(f+g) ? U(f) + U(g), form the basis for proving this property by showing that the approximation errors can be controlled in a manner that respects addition.

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