Dynamic system, fixed point Given the dynamic system $u^{k+1}=g(u^{k})$, and a vector $v ∈ \mathbb{R}^n$ that satisfies $v=g(v)$, which is said to be a fixed point of the system. 
i) Suppose that the solution for a dynamic system converge, in other words, $\lim u^{k+1}=u^*$. Show that $u^*$ is a fixed point.
PS: Sorry for my English.

We have that $\lim_{k \rightarrow +\infty} u^{k+1} = u^*$. By definition, This means that there exists a positive integer $M$ such that $u^k = u^*$ for all $k > M$. Suppose to take $k > M$ and consider that:
$$u^{k+1} = g(u^k)$$
We know that $u^k = u^*$ and also $u^{k+1}=u^*$, and hence $u^{k+1} = g(u^*) = u^*$. Then $u^*$ is a fixed point. 
