Column stochastic matrix/Stable/Fact/Proof

For an arbitrary vector ${\displaystyle {}v\in V}$, we have

${\displaystyle {}{\begin{aligned}\mid \mid \!Mv\!\mid \mid _{\operatorname {sum} }&=\sum _{i=1}^{n}\vert {(Mv)_{i}}\vert \\&=\sum _{i=1}^{n}\vert {\sum _{j=1}^{n}a_{ij}v_{j}}\vert \\&\leq \sum _{i=1}^{n}{\left(\sum _{j=1}^{n}a_{ij}\vert {v_{j}}\vert \right)}\\&=\sum _{j=1}^{n}\vert {v_{j}}\vert {\left(\sum _{i=1}^{n}a_{ij}\right)}\\&=\sum _{j=1}^{n}\vert {v_{j}}\vert \\&=\mid \mid \!v\!\mid \mid _{\operatorname {sum} }.\end{aligned}}}$

Iterative application of this observation shows that fact  (2) is satisfied.