Sensitivity to the initial conditions: an example.

Question

A dynamical system $\{f,I\}$ is said to be sensitive to the initial conditions, if there exists a $\delta > 0$ s.t. : \begin{align} & \forall x \in I, \forall \epsilon>0 \ \ \ \exists k \in \mathbb{N} , z \in I \ \ t.c. \\ & |x-z|<\epsilon, \ \ \ |f^k(z)-f^k(x)| > \delta \end{align} This is the definition provided in my notes. Now let $\{f,I\} = \{x^2,(1,+\infty)\}$. Then they say, let $X_0, \hat{X_0} \in I $ s.t. $ |x_0-\hat{x}_0| = \epsilon$, then: $$ |x_k-\hat{x}_k| = |x_0^{2k}-\hat{x}_0^{2k}| = |x_0-\hat{x}_0|| x_0^{2k-1} + x_0^{2k-2}\hat{x}_0+ ... +\hat{x}_0^{2k-1}| \ge 2k\epsilon $$ And they conclude this system is sensitive to the initial conditions. But, if that is the case, what would be the $\delta$ of the definition? The lower bound they found depends on $\epsilon$, so I don't understand if this proof is ok, or I'm missing something.

Answer 1

I'll expand on @peanojr's answer in the comments. Notice that since $\varepsilon>0$ and natural $k$ has to be chosen given $\varepsilon$, we can choose $k \in \mathbb{N}$ such that $2k\varepsilon \geq \delta$ for any $\delta>0$. This is due to the fact that we can find $k \in \mathbb{N}$, s.t. $$k>\frac{\delta}{2\varepsilon}>0$$ by the archimedean property of the reals. To finish the above proof you would have to choose an arbitrary $\delta>0$ and give an argument for how the inequality would hold if $|x_0-\hat{x}_0|<\varepsilon$ (hint: you can choose $|x_0-\hat{x}_0|=\frac{\varepsilon}{2}).$