How do I prove that the rank of a matrix in reduced row echelon form is equal to the number of non-zero rows it has?
By definition of row echelon form, all the non-zero rows are linearly independent. (Think on all the non-zero leading coefficients which are "aligned" ).
So the non-zero rows form a basis for the row space (i.e the subspace spanned by all the rows). Therefore their number equals to the dimension of this space, as required.