Limits and Cauchy sequences of reals

Depends on

• The real numbers
• Order on the reals

Used by

• A function has no limit at c as soon as two sequences in A ∖ {c} tending to c give different limits of the values Corollary
• For a series of real numbers, unconditional convergence and absolute convergence are the same property Corollary
• Stolz-Cesaro, 0/0 form: if bₖ is strictly decreasing to 0, aₖ → 0, and the difference quotient converges, then aₖ/bₖ converges to the same value Corollary
• The a priori bound d(x^*, xₙ) ≤ qⁿ d(x₁,x₀)/(1-q) and the a posteriori bound d(x^*, xₙ₊₁) ≤ q d(xₙ₊₁,xₙ)/(1-q) Corollary
• The Cauchy-sequence reals have the least-upper-bound property Corollary
• The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem Corollary
• The limit inferior is the least subsequential limit in overlineℝ Corollary
• (1-1) + (1-1) + … converges to 0 while ∑ₖ (-1)ᵏ diverges Counterexample
• ∏_j ≥ 0 (1 + (-1)ʲ/√j+2) has partial products tending to 0 although ∑_j ≥ 0 (-1)ʲ/√j+2 converges Counterexample
• ∑ k^-1/2 diverges and ∑ k⁻² converges, and both have root limit exactly 1 Counterexample
• 1/4 lies in the Cantor set and is the endpoint of no removed interval, so the endpoints do not exhaust it Counterexample
• A nonnegative non-monotone sequence for which ∑ aₖ and ∑ 2ᵏ a_2ᵏ behave differently Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in overlineℝ Counterexample
• A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one Counterexample
• aₖ = (-1)ᵏ, bₖ = k have aₖ/bₖ → 0 while the difference quotient oscillates, so Stolz-Cesaro has no converse Counterexample
• Continuous fₙ → 0 pointwise on [0,1] with ∫₀¹ fₙ = 1 for every n Counterexample
• For the Dirichlet function every uniform partition with rational tags gives Riemann sum 1, so the sums converge along that sequence of tagged partitions although the function is not integrable: the mesh condition of the Riemann definition quantifies over all tagged partitions and cannot be weakened to one sequence Counterexample
• Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence Counterexample
• ℝ is the union of a meager set and a set of measure zero, so smallness of category and smallness of measure are independent notions Counterexample
• The Cauchy product of ∑_k ≥ 0 (-1)ᵏ/√k+1 with itself has |cₙ| ≥ 1 for every n, so it diverges Counterexample
• The double sequence (m+1)/(m+n+2) has unequal iterated limits Counterexample
• The indicator of ℚ has a limit at no point of ℝ Counterexample
• The sequence 1, 1, 2, 1, 3, 1, 4, … is unbounded and has a convergent subsequence Counterexample
• The truncated decimal approximations of √2 form a Cauchy sequence of rationals with no rational limit Counterexample
• Two series with aₖ ≤ bₖ for all k, ∑ bₖ convergent and ∑ aₖ divergent, when the terms may be negative Counterexample
• With aⱼ = (-1)ʲ/√j+1 convergent and bⱼ = (-1)ʲ bounded but not monotone, ∑ aⱼ bⱼ = ∑ 1/√j+1 diverges Counterexample
• With aₖ/bₖ → 0, convergence of ∑ aₖ does not give convergence of ∑ bₖ Counterexample
• xₖ = √k has xₖ₊₁ - xₖ → 0 and is not Cauchy Counterexample
• xₖ₊₁ = xₖ + 1/xₖ from x₁ = 1 has strictly decreasing consecutive gaps and diverges, so no uniform c < 1 exists Counterexample
• ψ(1/x) has no limit at 0: two sequences tending to 0 give values constantly 0 and constantly 1/2 Counterexample
• A summability (Toeplitz) matrix, the transformed sequence yₙ = ∑ₖ c_n,k xₖ, and regularity Definition
• Absolutely convergent and conditionally convergent series, and the general starting index Definition
• Cauchy sequence in a metric space Definition
• Convergence in overlineℝ and the extended subsequential limit set: L ∈ overlineℝ is an extended subsequential limit when some subsequence converges to L, or diverges to L = ±∞ Definition
• Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ Definition
• Divergence to +∞ and to -∞ Definition
• Infinite products: partial products, and convergence to a nonzero limit after finitely many vanishing factors Definition
• Open cover, subcover, compact subset of ℝ (every open cover has a finite subcover), and sequentially compact subset Definition
• Pointwise convergence of a sequence of real functions, and the Baire class one functions as the pointwise limits of sequences of continuous functions Definition
• Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions Definition

…and 137 more results.

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Direct dependencies and their dependencies through the next three levels: 15 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §6.1 (standard reference, not scraped)
• Cauchy sequence (Wikipedia) (standard reference, not scraped)