generalization and the Poincaré and Brouwer theorem

Question

Theorem:
There exists a continuous vector field without singularities on the sphere $S^{n}$ , if only if , $n$ is odd.

I'm really stuck on how to prove this.

Answer 1

Neither are generally true.

If $\frac ab = -\frac 12$, then the right hand side is

$$-\left\lfloor-\frac ab\right\rfloor - 1 = -\left\lfloor\frac 12\right\rfloor - 1 = -0-1 = -1$$

For the first, as a counter example, consider $x = 0$. Then

$$\lfloor x \rfloor = 0 \ne -1$$

For the second, as a counter example, consider $x = -42$. Then

$$\lfloor x \rfloor = -42 \ne -1$$