Exercise 2.8.3

Question

Define
$$
    t_{mn}=  \sum^m_{i=1}\sum^n_{j=1}|a_{ij}|
$$
\begin{enumerate}[label=(\alph*)] 
    \item Prove that $(t_{nn})$ converges.
    \item Now, use the fact that $(t_{nn})$ is a Cauchy sequence to argue that $(s_{nn})$ converges.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
    \item Note that $t_{nn}$ is monotone increasing; moreover
    $$
         \sum^\infty_{i=1}\sum^\infty_{j=1}|a_{ij}|
    \geq \sum^\infty_{i=1}\sum^n_{j=1}     |a_{ij}|
    \geq \sum^n_{i=1}     \sum^n_{j=1}     |a_{ij}| = t_{nn}
    $$
    and therefore $t_{nn}$ is bounded; by the Monotone Convergence Theorem $t_{nn}$ converges.

\item Since $(t_{nn})$ is a Cauchy sequence, for any $\epsilon > 0$ there exists $N$ such that if $p>q>N$,
     $$
\begin{aligned}
    \epsilon > \left| t_{pp} - t_{qq} \right| &= \left| \sum^p_{i=1} \sum^p_{j=1} |a_{ij}| - \sum^q_{i=1} \sum^q_{j=1} |a_{ij}| \right| \
    &= \left| \sum^p_{i=q+1} \sum^p_{j=1} |a_{ij}| + \sum^q_{i=1} \sum^p_{j=q+1} |a_{ij}|  + \sum^q_{i=1} \sum^q_{j=1} |a_{ij}| - \sum^q_{i=1} \sum^q_{j=1} |a_{ij}| \right| \
    &= \left| \sum^p_{i=q+1} \sum^p_{j=1} |a_{ij}| + \sum^q_{i=1} \sum^p_{j=q+1} |a_{ij}|  \right| \
    &\geq \left|\sum^p_{i=q+1} \sum^p_{j=1} a_{ij}  + \sum^q_{i=1} \sum^p_{j=q+1} a_{ij} \right| \
    &= \left| \sum^p_{i=q+1} \sum^p_{j=1} a_{ij} + \sum^q_{i=1} \sum^p_{j=q+1} a_{ij}  + \sum^q_{i=1} \sum^q_{j=1} a_{ij} - \sum^q_{i=1} \sum^q_{j=1} a_{ij} \right|\
    &= \left|\sum^p_{i=1} \sum^p_{j=1} a_{ij} - \sum^q_{i=1} \sum^q_{j=1} a_{ij} \right| \
    &= \left|s_{pp} - s_{qq}\right|
\end{aligned}
$$

and therefore $s_{nn}$ is also a Cauchy sequence and thus converges.
 \end{enumerate}

Exercise 2.8.3

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Exercise 2.8.3

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