Given the differential equation $ dy/dx = f(x,y)$ with initial condition $y(x_0) = y_0$. Let $f$ be a continuous function in $x$ and $y$ and Lipschitz-continuous in $y$ with Lipschitz constant L. Let F be a mapping on the space $C^0(I)$ of continuous function $u : I \rightarrow \mathbb{R}.$ $I = [x_0,M]$.
$$Fu(x) = y_0 + \int_{x_0}^x f(s,u(s))ds. $$
Let for $a \geq 0$ the norm $|| \cdot||_\alpha$ be given on $C^0(I)$ by
$$||u||_\alpha = \max\limits_{x \in I} |u(x)e^{-\alpha x}|, \ \ \ \ u \in C^0(I)$$
How can I prove that this Initial Value Problem has a unique solution on $I$? Doesn't this follow directly from the fact that the function $f$ is Lipschitz-continuous?