Let $f\in C^1(\mathbb{R}^2,\mathbb{R})$ s.t. \begin{equation} \frac{\partial f}{\partial y}(t,x)\leq 0 \;\; \forall (t,x)\in \mathbb{R}^2, \;\; f(t,0)=0\;\; \forall t\in \mathbb{R}. \end{equation} Let us consider the Cauchy problem \begin{equation} \begin{cases} y'=f(t,y)\\ y(0)=x \end{cases} \end{equation} 1). Prove that, for all $x\in \mathbb{R}$, the solution $y_x$ to the problem is well defined on $[0,\infty[$.
2). Prove that the set $\{y_x(\cdot), x\in[0,1]\}$ is compact in $C([0,\infty[)$.
I am still in trouble with the first point: I think the local existence comes from the Cauchy-Lipschitz theorem and to extend it on the right I have tried to use comparison theorems (since $z=x$ is a supersolution it is an upper-bound and $z=0$ is a lower-bound so that for $x>0$ the statement should be proved) but I can't prove it for $x<0$.