α ⇒There exists N ∈ N such that ak &lt

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NOTES ON lim sup AND lim inf

Here, I provide a detailed explanation on the consequences of lim supka_k < α and lim supka_k > α. There are, of course, analogous statements for lim inf.

Proposition 0.1.

lim sup

k

a_k < α ⇒There exists N ∈ N such that ak < α ∀k ≥ N.

Proof. Since lim supka_k < α, there exists β < α so that lim supka_k < β < α. It is then sucient to show that ak is eventually ≤ β. That is, there is N ∈ N so that ak ≤ β ∀k ≥ N. If not, for each n ∈ N, there exists akn so that akn ≥ β. All these akn's form a subsequence of {ak}. Since lim supka_k ≤ α, for large enough n, we have

β ≤ a_k_n ≤ α

and without loss of generality we may assume the bounds are true for all n. By Bolzano-Weirstrass Theorem, akn (and so ak) has a convergent subsequence whose limit is no less than β. This implies that lim supka_k ≥ β, which contradicts the fact lim supka_k < β.

The converse of the proposition is false. Take, for example, ak = 1 − ¹_k. Then a_k < 1 ∀k, but lim supka_k= lim_ka_k = 1 ≮ 1. The partial converse of Proposition 0.1 is the following:

Proposition 0.2.

lim sup

k

a_k > α ⇒ There exists a subsequence {ak⁰} ⊂ {a_k} such that ak⁰ > α ∀k⁰. Proof. Pick β so that α < β < lim supka_k. By denition of lim sup, there is a convergent subsequence {ak⁰} ⊂ {a_k} so that ak⁰ → a ∈ (β, lim sup_ka_k) as k⁰ → ∞. Pick small enough so that β − > α. Since ak⁰ → a, there exists N ∈ N so that a^k⁰ > a − > β − > α for all k⁰ ≥ N. The subsequence {ak⁰}k⁰≥N

is the desired subsequence.

The analogous statements for lim inf, which are proved vice versa, are the followings.

Proposition 0.3.

lim inf

k a_k> α ⇒There exists N ∈ N such that ak> α ∀k ≥ N.

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2 NOTES ON LIM SUP AND LIM INF

Proposition 0.4.

lim inf

k ak < α ⇒ There exists a subsequence {ak⁰} ⊂ {ak} such that ak⁰ < α ∀k⁰. Of course, all propositions above can be stated equivalently with their contra- positives.

Proposition 0.5.

There exists a subsequence {ak⁰} ⊂ {a_k} such that ak⁰ ≥ α ∀k⁰ ⇒ lim sup

k

a_k ≥ α.

Proposition 0.6.

There exists N ∈ N such that ak ≤ α ∀k ≥ N ⇒ lim sup

k

a_k ≤ α.

Proposition 0.7.

There exists a subsequence {ak⁰} ⊂ {a_k} such that ak⁰ ≤ α ∀k⁰ ⇒ lim inf

k a_k ≤ α.

Proposition 0.8.

There exists N ∈ N such that ak≥ α ∀k ≥ N ⇒ lim inf

k a_k≥ α.