TITLE: compact Riemann surface and branched cover
QUESTION [5 upvotes]: Show that a compact Riemann surface admits a branched cover of the
sphere with only simple branch points. 
I have this problem and it seems to me that the way is to use Riemann Roch theorem, but I do not know how to exactly apply!
Any help is welcome.

REPLY [2 votes]: Choose a branch set $D$ of size greater than or equal to the genus. Then by Riemann-Roch, the dimension of the space of meromorphic functions branching on $D$ with at worst simple poles is greater than or equal to $1$.