Convergence theorem1

$Monotonically increasing, decreasing$

{s_{n}}

is monotonically increasing:

\forall n \geq 1 (s_{n} \leq s_{n + 1})

{s_{n}}

is monotonically decreasing:

\forall n \geq 1 (s_{n} \geq s_{n + 1})

{s_{n}}

is monotonically increasing

\Rightarrow {s_{n}}

is a divergent sequence

{s_{n}}

is monotonically decresing

\Rightarrow {s_{n}}

is a divergent sequence

α = sup X \Leftrightarrow

1.\forall x \in X (x \leq α) 2.\forall ε > 0, \exists x \in X, (x > α - ε)

β = in f X \Leftrightarrow

1.\forall x \in X, (x \geq β) 2.\forall ε > 0, \exists x \in X, (x < β + ε)

X \neq = \emptyset

is bounded to the above

\Rightarrow sup X

exists

X \neq = \emptyset

is bounded to the below

\Rightarrow in f X

exists

Subsequence of

{a_{n}}_{n = 1}^{\infty}

: {a_{n_{k}}}_{k = 1}^{\infty}

{a_{n}}_{n = 1}^{\infty}

is bounded sequence

\Rightarrow

convergent subsequence of

{a_{n}}_{n = 1}^{\infty}

exists

{a_{n}}_{n = 1}^{\infty}

is bounded

\Leftrightarrow {a_{n}}_{n = 1}^{\infty}

is a Cauchy sequence

Expected $