sequence limit

a sequence $\text{[math]}$ converges to $\text{[math]}$ if for all $\text{[math]}$ there exists some $\text{[math]}$ such that $\text{[math]}$ for all $\text{[math]}$ . when this happens, $\text{[math]}$ is called the limit of $\text{[math]}$ and we write $\text{[math]}$ .
[cite:;taken from @analysis_cummings_2019 chapter 3.3 convergent sequences; definition 3.7]

stuff from college

the limit of a sequence is denoted as
$\text{[math]}$ if for every $\text{[math]}$ there exists $\text{[math]}$ such that for every $\text{[math]}$ we know $\text{[math]}$

$\text{[math]}$ to find this limit we start with $\text{[math]}$ which we need to show is smaller than $\text{[math]}$ :
$\text{[math]}$

sequence limit table

$\text{[math]}$