Dominant weight as a positive combination of simple roots Let G be a semisimple algebraic group. 
I can see (geometrically) why every dominant weight has to be a non-negative combination of simple roots (and if it strictly dominant then it has to be a positive combination). I assume that the coefficients are allowed to be in $\mathbb{Q}$.
Does anybody have an algebraic proof?

To show the non-negativity. Write a dominant character as the sum of negative linear combination of some of the simple roots and non-negative of the others.
Now, taking scalar product with the negative part, gives the deusired result. 
