Sequences

A list of numbers written in a definite order

A sequence can be defined as a function whose domain is the set of positive integers

We deal with infinite sequences

{a₁, a₂, a₃, ...}

{a_n}

Sequences can be defined by giving a formula for the nth term.

For example:  $a_n=\frac{n}{n+1}$  

The n does not have to start at 1

Limit Laws hold for the limits of sequences

The Squeeze Theorem can be adapted for sequences.

A sequence is considered bounded if it bounded both above AND below.

For instance, the sequence $a_n=n$  is bounded below since  $a_n>0$ but not above.

The sequence  $a_n=\frac{n}{n+1}$  is bounded because  $0<a_n<1$  for all  $n$  

Is every bounded sequence convergent?

Is every monotonic sequence convergent?

Turns out that a sequence is convergent if it is both bounded AND monotonic.