# how to prove the image of a loop is the entire space

Show that there exists a loop $\gamma:[0,1]\rightarrow S^2$ such that the image of $\gamma$ is the entire $S^2$

i cannot prove it in a exactly correctly way, could anyone help me?

(Since $S^2$ is homeomorphic to $\mathbb R^2\cup \infty$ you are well on your way. ..)
For example, you should be able to find a curve from $(0,1)$ to $\mathbb R^2$. Then for your loop, $\gamma$, extend by mapping $0$ and $1$ to the north pole (point at infinity)