I'll give some background before my question, you may skip that if you like:
Lets say that you have the unit circle $S^1$.
It can be proved that the map $p: \mathbb{R} \rightarrow S^1$, given by $p(x)=(\cos(2\pi x),\sin(2\pi x))$ is a covering map.
With a lot of work, the book ends up proving this statement:
Let $p: E \rightarrow B$ be a covering map, let $p(e_0)=b_0$. If E is simply connected, then the lifting correspondence
$\phi: \pi_1(B, b_0) \rightarrow p^{-1}(b_0)$ is bijective.
The function $\phi$ is defined like this: If f is a loop in B based at $b_0$ it has a unique lifting to E, $\bar{f}$ that starts in $e_0$, $\phi([f])$ is defined to be the point in E, that is the end of this path.(So even though we start with a loop, the lifting may not be a loop).It can be shown that this function is well-defined.
Then it ends up with this theorem:
The fundamental group of $S^1$ is isomorphic to the additive group of integers.
QUESTION:
Later in the book. It is stated that for instance the loop $[0,1] \rightarrow (\cos(2\pi x),\sin(2\pi x))$, corresponds to the number 1 in the group $\mathbb{Z}$. And the loop $[0,n]\rightarrow (\cos2\pi x,\sin 2\pi x)$ corresponds to the number n in this group.
This seems very natural. But I can't see how this is proved to be correct? We know that $p^{-1}((1,0))$ is the integers. But lets say I have the loop given by $[0,n] \rightarrow (\cos(2\pi x,\sin 2\pi x))$, or written maybe like $[0,1] \rightarrow (\cos(2\pi n x,\sin 2\pi n x))$, if this loop should correspong to the number n in $\mathbb{Z}$, then the lifting of this path to $\mathbb{R}$ should end in n? How is this proved?
This is proved by observing that if $f : [0,1] \to S^1$ is given by the formula $$f(x) = (\cos(2 \pi n x), \sin(2 \pi n x)) $$ then the unique lifting of $f$ which starts at $0$ is given by the formula $$\tilde f(x) = nx $$
One way to discover this, perhaps, is to ask yourself: what function starting at $0$, when composed with $p(x)=(\cos(2 \pi x), \sin(2 \pi x)$ yields $f(x) = (\cos(2 \pi nx), \sin(2 \pi n))$?