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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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The absolute value of the limit, and the limit of the absolute value

Example

Let $x_{k} : = (k + 1)^{- 1} - 1$ . Then $x_{k} \to - 1$ , and $∣ x_{k} ∣ \to 1 = ∣ - 1 ∣$ (The absolute value is compatible with limits, Limits and Cauchy sequences of reals). Since $∣ x_{k} ∣ = 1 - (k + 1)^{- 1}$ converges to $1$ while $x_{k}$ converges to $- 1 \neq 1$ , the magnitude of the limit is recovered from the absolute values, but its sign is not. In particular, $∣ x_{k} ∣ \to ∣ 1 ∣$ does not imply $x_{k} \to 1$ : the implication of The absolute value is compatible with limits cannot be reversed for the candidate limit $1$ .

Facts & Assumptions

Given: The sequences $x_{k} : = (k + 1)^{- 1} - 1$ and $u_{k} : = (k + 1)^{- 1}$ , where $k + 1 = σ (k)$ is read in $R$ through the canonical-natural embedding (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[F1]

$(u_{k})$ converges to $0$ and every $u_{k}$ is positive (The sequence $1 / (k + 1)$ converges to $0$ ).

[F2]

Sum clause of the algebra of limits: if $(p_{k})$ converges to $p$ and $(q_{k})$ converges to $q$ , then $p_{k} + q_{k}$ converges to $p + q$ (Algebra of limits: sums, scalar multiples, products and quotients).

[F3]

Absolute values of limits: if $(z_{k})$ converges to $z$ then $(∣ z_{k} ∣)$ converges to $∣ z ∣$ (The absolute value is compatible with limits).

[F4]

Uniqueness of limits: if $(z_{k})$ converges to $z$ and to $z^{'}$ , then $z = z^{'}$ (A sequence has at most one limit).

[F5]

Absolute value: $∣ u ∣ = u$ if $u \geq 0$ and $∣ u ∣ = - u$ if $u < 0$ ; so $∣ 0 ∣ = 0$ and $∣ - 1 ∣ = - (- 1) = 1$ because $- 1 < 0$ (Absolute value in an ordered field).

[F6]

Reciprocals against $1$ : for $r > 0$ , $r \geq 1 ⟺ 0 < r^{- 1} \leq 1$ (Reciprocals and order: $1 / r$ against $1$ ).

[F7]

$0 < 1$ in $R$ (The multiplicative identity is positive).

[F8]

Translation invariance: $a < b$ implies $a + c < b + c$ (Order is preserved by adding a constant and by adding inequalities).

[F9]

Convergence: $(z_{k})$ converges to $z$ when for every rational $ε > 0$ there is $K$ with $∣ z_{k} - z ∣ < ε$ for all $k \geq K$ (Limits and Cauchy sequences of reals).

[F10]

Field arithmetic: $u - v = u + (- v)$ , $- (u - v) = v - u$ , and $- (- 1) = 1$ (Field).

[F11]

Trichotomy: for all reals $u, v$ exactly one of $u < v$ , $u = v$ , $v < u$ holds (Ordered field, Order on the reals).

Verification

technique · direct

1.1givenF1

The sequence $(u_{k})$ converges to $0$ with every term positive.

1.2F5F9

For every real $c$ the constant sequence $c$ converges to $c$ : given a rational $ε > 0$ and any index $k$ , $∣ c - c ∣ = ∣ 0 ∣ = 0 < ε$ by [F5], so the threshold $K = 0$ witnesses the condition of [F9].

2.1step 1.1step 1.2F2

The sum $x_{k} = u_{k} + (- 1)$ converges to $0 + (- 1) = - 1$ by the sum clause [F2] applied to step 1.1 and to the constant sequence $- 1$ of step 1.2.

2.2step 1.1F1F5F6F10F11

For every $k$ one has $0 < u_{k} \leq 1$ : positivity is [F1], and $u_{k} \leq 1$ is the case $r = k + 1 \geq 1$ of [F6]. Hence $x_{k} = u_{k} - 1 \leq 0$ . If $x_{k} < 0$ , the negative clause of [F5] gives $∣ x_{k} ∣ = - x_{k}$ ; if $x_{k} = 0$ , both sides are $0$ . Thus $∣ x_{k} ∣ = - x_{k} = 1 - u_{k}$ at every index, including $k = 0$ .

3.1step 2.1F3F5F7F8F10

Hence $∣ x_{k} ∣ \to ∣ - 1 ∣ = 1$ : step 2.1 and [F3] give $∣ x_{k} ∣ \to ∣ - 1 ∣$ , and $∣ - 1 ∣ = 1$ because $- 1 < 0$ by [F7] and [F8], so [F5] gives $∣ - 1 ∣ = - (- 1) = 1$ by [F10].

4.1step 2.1step 3.1F3F4F5F7F8F11∎

Consider the converse of [F3] for the candidate limit $c = 1$ : it would assert that $∣ x_{k} ∣ \to ∣ 1 ∣ = 1$ implies $x_{k} \to 1$ . Its hypothesis holds by step 3.1, but its conclusion would contradict $x_{k} \to - 1$ from step 2.1, since uniqueness [F4] would force $- 1 = 1$ , whereas $- 1 < 0 < 1$ by [F7] and [F8]. Thus this witness refutes the converse for the nonzero candidate $1$ , while satisfying the forward implication for its actual limit $- 1$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

John K. Hunter, An Introduction to Real Analysis (complete author-hosted lecture notes, 307 pp.) (standard reference, not scraped)
Jiri Lebl, Basic Analysis I: Introduction to Real Analysis, Volume I (author-hosted complete book PDF) (standard reference, not scraped)