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A bounded truncation function need not have an improper limit
Example
Define on for every nonnegative integer . Its truncation primitive is bounded, but diverges.
Facts & Assumptions
Given: The alternating unit-step function .
Proper integrals add across the integer subintervals (For : is integrable on if and only if it is integrable on and on , and then ; with the oriented form for arbitrary ).
The alternating sequence has partial sums alternating between two values (The even and odd index maps and the alternating sequence: strictly increasing with their disjoint union, and the unique with , , which satisfies , and ).
The bounded-primitive criterion requires a nonnegative integrand (A nonnegative improper integral converges iff its truncated integrals are bounded).
Verification
For a positive integer , additivity gives the displayed sum, whose parity values follow from the alternating sequence. [L1, L2] which equals one for odd and zero for even by [L2]. Thus the integer truncations are bounded but have no limit. [L1, L2] If , the remaining integral has absolute value at most one, so the full truncation primitive is bounded as asserted.
An improper limit would restrict to the same limit along all integer truncations, contradicting step 1.1. The example changes sign, so it does not satisfy the nonnegativity hypothesis in [L3] and shows that hypothesis cannot be deleted. [L3, step 1.1].
Depends on
- Improper integrals over unbounded intervals
- A nonnegative improper integral converges iff its truncated integrals are bounded
- For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then $\int_a^b f = \int_a^c f + \int_c^b f$; with the oriented form for arbitrary $a,b,c$
- The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $\mathbb{N}$ their disjoint union, and the unique $(s_k)$ with $s_0 = 1$, $s_{\sigma(k)} = -s_k$, which satisfies $|s_k| = 1$, $s \circ e \equiv 1$ and $s \circ o \equiv -1$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
Used by
Nothing in the library uses this result yet.
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Sources
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)