# 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?