Let $F:\mathbb{R}\times\mathbb{R}^d\to\mathbb{R}^d$ be a continuous and Lipschitz in the second variable function, and consider the IVP:
$x´=F(t,x)$
$x(t_0)=x_0$.
If $\gamma:(w^-,w^+)\to\mathbb{R}^d $ is a maximal solution of the IVP, then $(w^-,w^+)=\mathbb{R}$.
[my achievement so far]
I wanted to show it by contradiction, so If we assume that $w^+<\infty$, as $(t,\gamma(t))\to\partial(\mathbb{R}\times\mathbb{R}^d)$ whenever $t\to w^+$, we would have $\lim_{t\to w^+}|\gamma(t)|=\infty$.
But why is this a contradiction?