ordinary differential equations – Let $f$ be a continuous Lipschitz function, then exists unique solution for $x’=f(t,x)$ with $x(t_0)=x_0$ for all $tin mathbb{R}$


Let $f:mathbb{R}timesmathbb{R}^nrightarrow mathbb{R}^n$ be continuous and Lipschitz. How do I prove that there exists unique solution of the IVP $x’=f(t,x),quad x(t_0)=x_0$

for all $tin mathbb{R}$.

I tried to use the Banach fixed-point theorem, but I think that $M=C(mathbb{R},mathbb{R}^n)$ is not a complete metric space whit the supremum norm. The reason I tried this is to define $T:Mrightarrow M$ as $T(z)(t)=x_0+int_{t_0}^{t} f(s,z(s)),ds$ and prove that there exist a fixed point using the following inequality

$$|T(x)(t)-T(y)(t)|leq frac{L^n lvert t-t_0rvert^n}{n!}d(x,y),$$

where $L$ is the Lipschitz constant for $f$ and $d(x,y)=suplimits_{tin mathbb{R}}|x(t)-y(t)|$.

Can somebody give me a hand?