Linear combination of simple function is a simple function

Question

Let $f_i= \sum_{j=1}^m a_j\chi_{A_j}$ be a simple function. We have to show that finite linear cimbination of simple functions is a simple function. $$ \sum_{i=1}^n \alpha_i f_i = \sum_{i=1}^n \alpha_i \sum_{j=1}^m a_j\chi_{A_j} = \sum_{i=1}^n\sum_{j=1}^m \alpha_i a_j\chi_{A_j} $$ I guess it is almost finish but I do not see why this sum satisfies the definiction of simple function.