Let f:[a, b] →ℝ be any function. Suppose there is r ∈ℝ and for each n ∈ℕ, there are integrable f

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Let f:[a, b] →ℝ be any function. Suppose there is r ∈ℝ and...

Let $f:[a, b] \rightarrow \mathbb{R}$ be any function. Suppose there is $r \in \mathbb{R}$ and for each $n \in \mathbb{N}$, there are integrable functions $g_{n}, h_{n}:[a, b] \rightarrow \mathbb{R}$ with $g_{n} \leq f \leq h_{n}$ such that $\int_{a}^{b} g_{n}(x) d x \rightarrow r$ and $\int_{a}^{b} h_{n}(x) d x \rightarrow r$ as $n \rightarrow \infty$. Show that $f$ is integrable and the Riemann integral of $f$ is equal to $r$.

Let $f:[a, b] \rightarrow \mathbb{R}$ be any function. Suppose there is $r \in \mathbb{R}$ and for each $n \in \mathbb{N}$, there are integrable functions $g_{n}, h_{n}:[a, b] \rightarrow \mathbb{R}$ with $g_{n} \leq f \leq h_{n}$
such that $\int_{a}^{b} g_{n}(x) d x \rightarrow r$ and $\int_{a}^{b} h_{n}(x) d x \rightarrow r$ as $n \rightarrow \infty$. Show that $f$ is integrable and the Riemann integral of $f$ is equal to $r$.

Key Concepts

Approximation by Integrable Functions

The method of approximating a function by integrable functions is a powerful technique in analysis. By finding sequences of functions that are easier to integrate and that converge to the target function from above and below, one can leverage their integrability properties to establish the integrability of the target function. This process is central to many proofs in integration theory, as it transfers the desirable properties of well-behaved functions to more complex ones.

Squeeze Theorem in the Context of Integration

The Squeeze (or Sandwich) Theorem is a fundamental analytical tool that states if a function is bounded from above and below by two other functions which converge to the same limit, then the function itself must also converge to that limit. In the context of Riemann integration, if a function is bounded by integrable functions whose integrals converge to the same value, it implies that the integral of the function must also converge to that value, thereby confirming its integrability.

Darboux Sums

Darboux sums are an alternative formulation of Riemann integration using the supremum (upper sum) and infimum (lower sum) of the function over subintervals of a partition. These sums provide a useful approach to proving integrability by establishing that if the difference between the upper and lower sums can be made smaller than any given tolerance, then the function is integrable. This concept formalizes the idea of approximating an area under a curve from above and below.

Riemann Integrability

Riemann integrability refers to a function’s ability to be integrated in the Riemann sense; that is, a function defined on a closed interval is Riemann integrable if, roughly speaking, its upper and lower sums can be made arbitrarily close. This concept critically depends on the behavior of the function, particularly on how well it can be approximated by simpler functions where the discrepancy between over-estimates and under-estimates becomes negligible as the partition is refined.

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