In this video, at just after the 5 minute mark, the speaker says: "...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram are closed cofibrations, and there is a theorem that if you have a diagram where everything is a...

$A_{n\times n}$ is a matrix having each row sum $<1$ and its largest eigenvalue is also $<1$. I need to show $A^t\to 0,\text{ i.e } a^t_{ij}\to 0\forall i,j\text{ as } t\to\infty$ given that $0<a_{ij}<1$. Well, suppose $0<\lambda<1$ be the largest eigenvalue of $A$ then $A^tx=\lambda^tx\to 0\tex...