Sequences

A sequence $(a_n)$ is a function $a:\mathbb{N} \rightarrow \mathbb{R}$ that maps natural numbers $n \in \mathbb{N}$ to real numbers $a_n \in \mathbb{R}.$

A sequence $(a_n)$ is bounded by real numbers $A$ and $B$ if and only if $A \leq a_n \leq B \, \, \forall n.$ Equivalently, $(a_n)$ is bounded if and only if $\exists M \geq 0 : \lvert a_n\rvert \leq M \, \, \forall n.$

A sequence $(a_n)$ is monotonically increasing if and only if $a_{n+1} \geq a_n \, \, \forall n$ and monotonically decreasing if and only if $a_{n+1} \leq a_n \, \, \forall n.$ A sequence is monotonic if it is either monotonically increasing or monotonically decreasing.

A sequence $(a_n)$ converges to a real number $L$ if and only if for every $\varepsilon >0$ there exists $N \in \mathbb{N}$ such that $a_n$ is $\varepsilon$-close to $L$ for all $n \geq N.$

$$\forall \varepsilon > 0 \, \exists N \in \mathbb{N} : \lvert a_n-L \rvert \leq \varepsilon \, \forall n \geq N$$

Every convergent sequence is bounded and every bounded monotonic sequence is convergent. A sequence is divergent if it is not convergent.

A sequence $(a_n)$ is a Cauchy sequence if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that for all natural numbers $m, n \geq N,$ $a_m$ and $a_n$ are $\varepsilon$-close.

$$\forall \varepsilon > 0 \, \exists N \in \mathbb{N} : \lvert a_n-a_m \rvert \leq \varepsilon \, \forall m,n \geq N$$

A Cauchy sequence is a sequence whose terms become arbitrarily close to one another. A sequence of real numbers is convergent if and only if it is a Cauchy sequence. This holds because the real numbers are complete; it fails in the rational numbers, where the decimal truncations $1, 1.4, 1.41, 1.414, \ldots$ of $\sqrt{2}$ form a Cauchy sequence that converges to no rational number.