Endomorphism/Power sequence/Convergence/Eigenvector or 0/Fact/Proof

Proof

Let ${}w$ denote the limit vector. Then

${}\varphi ^{n+1}(v)=\varphi (\varphi ^{n}(v))\,$

holds for all ${}n\in \mathbb {N}$. Because of the continuity (according to fact) of ${}\varphi$, the limit and the mapping may be swapped (according to fact). That is,

${}w=\lim _{n\rightarrow \infty }\varphi ^{n}(v)=\lim _{n\rightarrow \infty }\varphi ^{n+1}(v)=\lim _{n\rightarrow \infty }\varphi (\varphi ^{n}(v))=\varphi {\left(\lim _{n\rightarrow \infty }\varphi ^{n}(v)\right)}=\varphi (w)\,.$

Therefore, ${}w$ is a fixed point of ${}\varphi$; that means, it is the zero vector, or an eigenvector for the eigenvalue ${}1$.

To fact