Doubly Indexed Array

linuxux

HELLO. GOT A QUICK QUESTION:

FROM THE TEXT:

[tex]|S_{mn} - S| <\ \in\ for\ all\ m,n \ge N.[/tex]

[tex]For\ the\ moment,\ consider\ m\ \in\ N\ to\ be\ fixed\ and\ write\ S_{mn}\ as:[/tex]

[tex]S_{mn}\ =\ \sum^{n}_{j=1}a_{1j}\ +\ \sum^{n}_{j=1}a_{2j}\ +\ ...\ +\ \sum^{n}_{j=1}a_{mj}[/tex]

[tex]Our\ hypothesis\ guarantees\ that\ for\ each\ fixed\ row,\ i,\ the\ series\ \sum^{\infty}_{j=1}\ converges\ absoloutely\ to\ some\ real\ number,\ r_i.[/tex]

HERES WHAT I HAVE TO DO:

[tex]Exercise\ 2.8.5.\ (a)\ Use\ the\ Algebraic\ Limit\ Theorem\ and\ the\ Order\ limit\ theorem\ to\ show\ that\ for\ all\ m\ge N\ |(r_{1},\ +\ r_{2},\ +\ ...\ +\ r_{m})\ -\ S| \le \in[/tex]

[tex]Conclude\ that\ the\ iterated\ sum\ \sum^{\infty}_{i=1}\sum^{\infty}_{j=1}\ converges\ to\ S[/tex]

Obviously the larger of an m you take r_m to means the closer you'll get to S, and since m belongs to N then the largest it can go is infinity, which would mean the double sum would indeed converge to S. Is that "enough" of a conclusion?