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Convergence depends only on the tail

Statement

Let $(x_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), let $x \in \mathbb{R}$ , and for $K \in \mathbb{N}$ let $x^{(K)}$ denote the $K$ -th tail, $x^{(K)}_j = x_{j+K}$ . The following are equivalent (Limits and Cauchy sequences of reals):

$(x_k)$ converges to $x$ ;
every tail $x^{(K)}$ converges to $x$ ;
some tail $x^{(K)}$ converges to $x$ .

The same three statements with "converges to $x$ " replaced by "is Cauchy" are likewise equivalent. Consequently, if two sequences agree from some index on, then either both converge to $x$ or neither does, and either both are Cauchy or neither is.

Facts & Assumptions

Given: A sequence $(x_k)$ of reals, a real $x$ , and for each $K \in \mathbb{N}$ the $K$ -th tail $x^{(K)}$ defined by $x^{(K)}_j := x_{j+K}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[L1]

Convergence and the Cauchy condition: $(z_k)$ converges to $z$ when for every rational $\varepsilon > 0$ there is $N$ with $|z_k - z| < \varepsilon$ for all $k \ge N$ , and $(z_k)$ is Cauchy when for every rational $\varepsilon > 0$ there is $N$ with $|z_k - z_l| < \varepsilon$ for all $k, l \ge N$ (Limits and Cauchy sequences of reals).

[L2]

Index arithmetic, first half: $j \ge N$ implies $j + K \ge N + K$ , and $N + K \ge N$ (Order on the natural numbers, Order is compatible with addition).

[L3]

Index arithmetic, second half: every $k \ge N + K$ has the form $k = j + K$ with $j \ge N$ . Indeed $N + K \le k$ gives $i$ with $(N+K) + i = k$ ; put $j := N + i$ , so that $j \ge N$ and $j + K = (N+i) + K = N + (i + K) = N + (K + i) = (N+K) + i = k$ (Order on the natural numbers, Addition is associative, Addition is commutative).

[L4]

The order on $\mathbb{N}$ is reflexive and transitive, so $k \ge N + K \ge N$ gives $k \ge N$ ; and $j + 0 = j$ for every $j$ ( $\le$ is a linear order on $\mathbb{N}$ , Order on the natural numbers, Addition of natural numbers).

Proof

technique · direct

1.1

(1) implies (2). Assume $(x_k)$ converges to $x$ , let $K \in \mathbb{N}$ be arbitrary and let $\varepsilon > 0$ be rational; take $N$ with $|x_k - x| < \varepsilon$ for all $k \ge N$ . For every $j \ge N$ we have $j + K \ge N + K \ge N$ , hence $|x^{(K)}_j - x| = |x_{j+K} - x| < \varepsilon$ ; so $x^{(K)}$ converges to $x$ .

assume-hypL1L2L4

1.2

(2) implies (3), in both families. The $0$ -th tail satisfies $x^{(0)}_j = x_{j+0} = x_j$ , so $(x_k)$ is itself one of its own tails; statement (2) quantifies over all tails, so applying it to this one already yields (3).

givenL4

1.3

(3) implies (1). Assume $x^{(K)}$ converges to $x$ for some fixed $K$ , and let $\varepsilon > 0$ be rational; take $N$ with $|x^{(K)}_j - x| < \varepsilon$ for all $j \ge N$ . Every $k \ge N + K$ is of the form $j + K$ with $j \ge N$ , so $|x_k - x| = |x^{(K)}_j - x| < \varepsilon$ ; taking $N + K$ as the threshold shows $(x_k)$ converges to $x$ .

assume-hypL1L3

1.4

The Cauchy version of (1) implies (2). Assume $(x_k)$ is Cauchy, fix $K$ and a rational $\varepsilon > 0$ , and take $N$ with $|x_k - x_l| < \varepsilon$ for all $k, l \ge N$ . For $i, j \ge N$ both $i + K \ge N$ and $j + K \ge N$ , so $|x^{(K)}_i - x^{(K)}_j| = |x_{i+K} - x_{j+K}| < \varepsilon$ ; so $x^{(K)}$ is Cauchy.

assume-hypL1L2L4

1.5

The Cauchy version of (3) implies (1). Assume $x^{(K)}$ is Cauchy and let $\varepsilon > 0$ be rational; take $N$ with $|x^{(K)}_i - x^{(K)}_j| < \varepsilon$ for all $i, j \ge N$ . Any $k, l \ge N + K$ are of the form $i + K$ and $j + K$ with $i, j \ge N$ , so $|x_k - x_l| < \varepsilon$ ; so $(x_k)$ is Cauchy.

assume-hypL1L3

2.1

In each of the two families the cycle (1) implies (2) implies (3) implies (1) is closed, so within each family the three statements are equivalent.

step 1.1step 1.2step 1.3step 1.4step 1.5

3.1

If $(x_k)$ and $(y_k)$ agree from index $K$ on, then $x^{(K)} = y^{(K)}$ as functions, so by the established equivalence $(x_k)$ converges to $x$ exactly when $y^{(K)}$ does, exactly when $(y_k)$ does, and the same chain applies to the Cauchy condition.

step 2.1given∎

Remarks

This is the lemma that makes the word "eventually" usable. Once it is known that finitely many initial terms are irrelevant, a hypothesis of the form " $P(k)$ for all $k$ " can everywhere be weakened to " $P(k)$ eventually" (Sequences of reals: bounded, eventually, frequently, tails, subsequences) at the cost of passing to a tail.
It also settles the bookkeeping question of where the indexing starts. A text that writes $(x_k)_{k \ge 1}$ is describing the first tail of a sequence in the sense used here, and the two have the same limits and the same Cauchy status.
Boundedness, by contrast, is not a tail property in the same trivial way: a tail of a bounded sequence is bounded, and a sequence with a bounded tail is bounded only because the finitely many omitted terms can be absorbed into the bound, which is exactly the argument of Every convergent sequence is bounded.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 12 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

J. K. Hunter, An Introduction to Real Analysis, Ch. 3 (standard reference, not scraped)
Limit of a sequence (Wikipedia) (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed., §6.1 (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)