Line bundles correspoding to a hyperplane

Question

Assume we have a smooth projective variety $X$ over a field and a hyperplane section $H$ on it. For each Weil divisor on $X$ you can construct a line bundle on $X$. For $H$ this line bundle which is denoted by $O(H)$ will be given as a sheaf by $O(H)(U)=\{f\in K| f+H\geq 0$ on $U \}$. So by this definition it is given as a sub sheaf of the constant sheaf given by $K$ which is the function field of $X$. What strikes me as strange is that you can give similar definitions for all $O(nH)$ and $O$ which is the trivial line bundle and by this definition it looks like that $O(nH)$ is a subsheaf of $O(mH)$ for $n<m$. Specially $O$ is a subsheaf of $O(H)$. Is this correct? or what exactly am I missing?