Definitions

Equivalence definitions of a subsequential limit

$L \in R$ is a subsequential limit of $(a_{n})_{n = 1}^{\infty}$
(d3.26) There exists subsequence $(a_{n_{k}})_{k = 1}^{\infty}$ such that $a_{n_{k}} \to L$
(3.27) $\forall ε > 0$ there are infinite $n$ values for which $∣ a_{n} - L ∣ < ε$
(3.27) $\forall ε > 0, \forall N \in N, \exists n > N, ∣ a_{n} - L ∣ < ε$
(3.27 geometrically) For every neighborhood of $L$ there are infinite terms of $(a_{n})$

Theorems

$\infty$ (or $- \infty$ ) is a subsequential limit of $(a_{n})_{n = 1}^{\infty}$ if there exists subsequence $(a_{n_{k}})_{k = 1}^{\infty}$ such that $a_{n_{k}} \to \infty$ (or $a_{n_{k}} \to - \infty$ )
(q3.38) $\infty$ is a subsequential limit of $(a_{n})$ if and only if $(a_{n})$ is not bounded above

The limit superior of $(a_{n})$
$n \to \infty \overline{lim} a_{n} = n \to \infty lim sup a_{n} := n \to \infty lim (m \geq n sup a_{m}) = n \to \infty lim (sup {a_{m} : m \geq n})$
The maximal subsequential limit of $(a_{n})$
The supremum of the set of all subsequential limits of $(a_{n})$

The limit inferior of $(a_{n})$
$n \to \infty \underline{lim} a_{n} = n \to \infty lim inf a_{n} := n \to \infty lim (m \geq n in f a_{m}) = n \to \infty lim (in f {a_{m} : m \geq n})$
The minimal subsequential limit of $(a_{n})$
The infimum of the set of all subsequential limits of $(a_{n})$

(3.41, 3.38) Existence - Each sequence has a limit superior and limit inferior
- if it’s bounded above, then $lim sup$ is finite, else is $\infty$
- if it’s bounded below, then $lim in f$ is finite, else is $- \infty$
(3.40)
- If $n \to \infty \overline{lim} a_{n} < c$ then for almost all $n$ then $a_{n} < c$
- If $c < n \to \infty \overline{lim} a_{n}$ then there are $\infty$ values of $n$ such that $c < a_{n}$
- $n \to \infty \overline{lim} (a_{n} + b_{n}) \leq n \to \infty \overline{lim} (a_{n}) + n \to \infty \overline{lim} (b_{n})$
- $n \to \infty \overline{lim} (a_{n} \cdot b_{n}) \leq n \to \infty \overline{lim} (a_{n}) \cdot n \to \infty \overline{lim} (b_{n})$ (if $a_{n}, b_{n} \geq 0$ )
- if $c > 0$ then $n \to \infty \overline{lim} (c a_{n}) = c n \to \infty \overline{lim} (a_{n})$
- if $a_{n} \leq b_{n}$ for each $n$ then $n \to \infty \overline{lim} a_{n} \leq n \to \infty \overline{lim} b_{n}$
(q3.56) $n \to \infty lim inf a_{n} = n \to \infty lim sup a_{n}$ if and only if $(a_{n})$ is convergent
(q3.59)
- $n \to \infty \overline{lim} a_{n} = - \infty ⟹ n \to \infty lim a_{n} = - \infty$
- $n \to \infty \underline{lim} a_{n} = \infty ⟹ n \to \infty lim a_{n} = \infty$
$n in f x_{n} \leq n \to \infty lim inf x_{n} \leq n \to \infty lim sup x_{n} \leq n sup x_{n}$
$n \to \infty \overline{lim} (a_{n} b_{n}) = n \to \infty lim a_{n} n \to \infty \overline{lim} (b_{n})$ (Kenneth, t12.1)