$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\FF}{\mathbb{F}} \renewcommand{\epsilon}{\varepsilon} % ALTERNATE VERSIONS % \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} % \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} % \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} % \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} % \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\uppersum}[1]{U_{#1}} \newcommand{\lowersum}[1]{L_{#1}} \newcommand{\upperint}[1]{U_{#1}} \newcommand{\lowerint}[1]{L_{#1}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} % extra auxiliary and additional topic/proof \newcommand{\extopic}{\bigstar} \newcommand{\auxtopic}{\blacklozenge} \newcommand{\additional}{\oplus} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\sampleset}[1]{\mathcal{S}_{#1}} \newcommand{\erf}{\operatorname{erf}} $$

11  \(\blacklozenge\) Limits of Limits

So far we have looked at sequences which depend on a single index \(n\). But there are many natural situations in mathematics where two or more indices come into play. We take a brief look at these here as the subtleties that arise when trying to compute their limits

Definition 11.1 (Double Sequence) A double sequence is an assignment of a value \(a_{mn}\) for \(m,n\in\NN\).

For example, \(a_{mn}=\frac{1}{m+n^2}\) is a double sequence. We can visualize a double sequence by writing not a list of numbers, but a 2 dimensional array. How should we define the limit of such an array?

Definition 11.2 (Limit of a Double Sequence)  

Example 11.1  

Definition 11.3 (Iterated Limit)  

Do we want to include information on double sequences \(a_{m,n}\) and taking limits in \(n\), \(m\) or both?

\[\lim_n\lim_m \frac{n}{n+m}=\lim_n\left(\lim_m \frac{n}{n+m}\right)=\lim_n 0=0\] \[\lim_m\lim_n\frac{n}{n+m} = \lim_m\left(\lim_n \frac{n}{n+m}\right)=\lim_m 1=1\]