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The limit inferior is the least subsequential limit in $\overline{R}$

Statement

Let $(x_{k})$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences). Then ${lim inf}_{k} x_{k} \in \overset{}{SL} (x)$ and ${lim inf}_{k} x_{k} \leq L$ for every $L \in \overset{}{SL} (x)$ (Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ , Convergence in $\overline{R}$ and the extended subsequential limit set: $L \in \overline{R}$ is an extended subsequential limit when some subsequence converges to $L$ , or diverges to $L = \pm \infty$ ).

So the extended subsequential limit set of any real sequence has a least element as well as a greatest one, and the two are ${lim inf}_{k} x_{k}$ and ${lim sup}_{k} x_{k}$ respectively (The limit superior is itself a subsequential limit in $\overline{R}$ and is the greatest one). Every extended subsequential limit lies between them.

Facts & Assumptions

Given: A sequence $(x_{k})$ of reals, and its reflection $y_{k} : = - x_{k}$ .

[L1]

Reflection on $\overline{R}$ : $a \mapsto - a$ satisfies $- (- a) = a$ and $a \leq b$ if and only if $- b \leq - a$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined).

[L2]

${lim sup}_{k} (- x_{k}) = - {lim inf}_{k} x_{k}$ ( $lim sup (- x_{k}) = - lim inf (x_{k})$ , with the reflection of $\overline{R}$ exchanging $\pm \infty$ ).

[L3]

For every real sequence the extended subsequential limit set is nonempty and has greatest element the limit superior (The limit superior is itself a subsequential limit in $\overline{R}$ and is the greatest one).

[L4]

Convergence in $\overline{R}$ , subsequences and the set $\overset{}{SL}$ (Convergence in $\overline{R}$ and the extended subsequential limit set: $L \in \overline{R}$ is an extended subsequential limit when some subsequence converges to $L$ , or diverges to $L = \pm \infty$ , Subsequential limit of a real sequence, and the subsequential limit set, Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals).

[L5]

Scalar multiples of convergent sequences: $z_{j} \to z$ in $R$ implies $c z_{j} \to c z$ (Algebra of limits: sums, scalar multiples, products and quotients).

[L6]

Divergence to $\pm \infty$ , and order reversal: $z_{j} > M$ is equivalent to $- z_{j} < - M$ , and $M$ runs over all reals exactly when $- M$ does (Divergence to $+ \infty$ and to $- \infty$ , Order is preserved by adding a constant and by adding inequalities).

[L7]

Limit superior and limit inferior of a real sequence (Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ ).

Proof

technique · direct

1.1

Put $y_{k} : = - x_{k}$ , a sequence of reals; then $- y_{k} = x_{k}$ for every $k$ , by the involution property of the reflection.

givenL1L4

1.2

Let $L \in \overline{R}$ and let $n : N \to N$ be strictly increasing with $(x_{n_{j}})$ converging to $L$ in $\overline{R}$ .

givenL4

1.3

By [L3] applied to the sequence $(y_{k})$ , the set $\overset{}{SL} (y)$ is nonempty and has greatest element $N_{0} : = {lim sup}_{k} y_{k}$ , and $N_{0} = - {lim inf}_{k} x_{k}$ by [L2].

givenL2L3L7

2.1

The reflected subsequence $(y_{n_{j}}) = (- x_{n_{j}})$ converges to $- L$ in $\overline{R}$ . If $L$ is real this is the scalar rule with $c = - 1$ . If $L = + \infty$ then for every real $M$ there is $J$ with $x_{n_{j}} > M$ for all $j \geq J$ , hence $y_{n_{j}} < - M$ for all such $j$ ; since $- M$ runs over all reals as $M$ does, $y_{n_{j}} \to - \infty = - L$ . If $L = - \infty$ the same argument with the inequalities exchanged gives $y_{n_{j}} \to + \infty = - L$ .

step 1.2L1L4L5L6

3.1

Hence $L \in \overset{}{SL} (x)$ implies $- L \in \overset{}{SL} (y)$ , the same index map serving. Applying that implication to the sequence $(y_{k})$ , whose reflection is $(x_{k})$ , gives conversely that $N \in \overset{}{SL} (y)$ implies $- N \in \overset{}{SL} (x)$ . So $\overset{}{SL} (x) = {- N : N \in \overset{}{SL} (y)}$ .

step 2.1step 1.1L1L4

4.1

Therefore $- N_{0} \in \overset{}{SL} (x)$ , and $- N_{0} = - (- {lim inf}_{k} x_{k}) = {lim inf}_{k} x_{k}$ ; and for any $L \in \overset{}{SL} (x)$ the element $- L$ lies in $\overset{}{SL} (y)$ , so $- L \leq N_{0}$ by maximality, whence ${lim inf}_{k} x_{k} = - N_{0} \leq L$ by order reversal. Thus ${lim inf}_{k} x_{k}$ is the least element of $\overset{}{SL} (x)$ .

step 3.1step 1.3L1L2∎

Remarks

Nothing is reconstructed. The subsequence realising ${lim inf}_{k} x_{k}$ is the one produced by The limit superior is itself a subsequential limit in $\overline{R}$ and is the greatest one for the reflected sequence, read back through $y \mapsto - y$ . That is the whole point of proving $lim sup (- x_{k}) = - lim inf (x_{k})$ , with the reflection of $\overline{R}$ exchanging $\pm \infty$ first: the recursion and the well-ordering argument are done once.
Combined with the greatest element, this brackets every subsequential limit. For any real sequence and any $L \in \overset{}{SL} (x)$ , ${lim inf}_{k} x_{k} \leq L \leq {lim sup}_{k} x_{k},$ which contains $lim inf x_{k} \leq lim sup x_{k}$ for every real sequence as the special case obtained by taking for $L$ either endpoint, both of which are in the set.
The real subsequential limit set inherits the statement only when the value is finite. If ${lim inf}_{k} x_{k}$ is a real number it is the least element of $SL (x)$ as well, since the two sets agree on $R$ (Convergence in $\overline{R}$ and the extended subsequential limit set: $L \in \overline{R}$ is an extended subsequential limit when some subsequence converges to $L$ , or diverges to $L = \pm \infty$ ). If it is $- \infty$ , then $SL (x)$ may have no least element at all, or be empty.

Depends on

Used by

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Sources

Limit superior and limit inferior (Wikipedia) (standard reference, not scraped)
Subsequential limit (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (3.17) (standard reference, not scraped)
N. Donaldson, Math 140A: Real Analysis notes (standard reference, not scraped)

The limit inferior is the least subsequential limit in R‾R