The strategy is to Taylor expand $F^n$ near $x^*$ and show that the linear part dominates for small displacements. Setting $y_n = x_n - x^*$, the linearisation gives $y_{n+1} = Ay_n + O(|y_n|^2)$ where $A = JF_{x^*}$. By induction, $F^n(x^* + y_0) = x^* + A^ny_0 + O(|y_0|^2)$ for $|y_0|$ small. If all $|\lambda_i| < 1$, then $\|A^n\| \leq C\rho^n$ for some $\rho < 1$ and $C > 0$ (by the spectral radius formula), so $|A^ny_0| \to 0$ geometrically and the nonlinear remainder is of smaller order, giving asymptotic stability. If $|\lambda_i| > 1$ for some $i$, then $A^n$ grows exponentially in the corresponding eigendirection, and the linear term dominates for small $|y_0|$, so the fixed point is unstable. The non-hyperbolic case requires higher-order analysis because the linear and nonlinear terms are of comparable size on the centre subspace.