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?
Hint: think of the space-filling curve (which is a famous example in topology).
The Peano curve in particular, or space-filling curves in general, are examples of this type of phenomenon...
See if you can transfer these ideas to the sphere...
(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)