I am trying to prove the following Lemma:
Lemma: Suppose that the point $x^*$ is a fixed point of
$x(n + 1) = f(x(n))$ (1)
while also an asymptotically stable(unstable) fixed point with respect to
$g(x) = f^2(x) = f(f(x))$ (2)
Then $x^*$ is also asymptotically stable(unstable) fixed point of (1).
The definition from the material I am studying for being a fixed point $x^*$ of a function $f(x)$ is that applying the function $f$ onto a fixed point $x^*$ results in the same point(i.e. $f(x^*) = x^*$).
And the definition being used for a fixed point $x^*$ to be asymptotically stable with respect to a function $g(x)$ is that the fixed point $x^*$ satisfies these 2 conditions:
The fixed point $x^*$ is stable in that when given an $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x_0 - x^*| < \delta$ then $|g^n(x_0) - x^*| < \epsilon$ for all $n > 0$.
The fixed point is also attracting in that there exists a $\eta > 0$ such that if $|x_0 - x^*| < \eta$ then $\lim\limits_{n \rightarrow \infty} x(n) = x^*$.
I am having trouble understanding and proving the above lemma. What I have so far is that if the assumptions hold for $x^*$ to be a fixed point of $f$, then $f(x^*) = x^*$ so $f(x^*) = f(f(x^*)) = f^2(x^*) = g(x^*)$. Would it then be suffice to conclude that applying $f(x^*)$ is the same as applying $g(x^*)$ and so the 2 conditions for $x^*$ to be asymptotically stable are met(due to $x^*$ being asymptotically stable with respect to $g$) and so $x^*$ is asymptotically stable with respect to $f$? I am not convinced a direct proof such as the one I have used includes all the necessary details and it seems likely that the proof of this lemma is not as straightforward or simple as it would seem. Any help with my confusion would be appreciated, thanks.