What's the easiest way to see this?
I can imagine a proof for $n=2^k$ since for some $P \in E(K)$ you can just move a line intersecting P round the curve till it's tangent, then that point, say $Q \in E(K)$ would be such that $2Q=P$ and induction would get the rest.
Struggling to see the proof for n not a power of 2 however.
The claim is true if $K$ is algebraically closed, and it follows from a more general fact about algebraic curves: every map $\phi\colon X\to Y$ of algebraic curves over an algebraically closed field is either constant or surjective. So you just need to show that multiplication by $n$ is not constant, which is a very easy exercise.