Prove that there are no fixed points in a non-autonomous system

Question

Consider the non-autonomous dynamical system

$$ \dot x = f(x,t) $$

with $x \in \mathbb{R}^n$. This may be converted to an autonomous system of dimension $n+1$ with $t = x_{n+1}$ and $\dot x_{n+1} = 1$.

Question: how can one prove that the new system has no fixed points?

Answer 1

Hint: A fixed point $y_0 \in \mathbb R^{n+1}$ of the new system $\dot y = F(y)$ with $F(y) = \binom{f((y_1, \ldots, y_n), y_{n+1})}{1}$ has $F(y_0) = 0$.