Rounding error?

Consider the following scenario.

A group of $n$ friends together gather an amount of money, contributing $a_1, a_2, \ldots, a_n$ of the total. Collectively, they manage to buy $m$ horses. They now want to distribute these horses according to the money they contribute. Normally, the $i$ -th person should get $m a_i$ horses, but since horses cannot be divided, these numbers need to be rounded. The $i$ -th person starts by getting $\lfloor m a_i \rfloor$ horses. At the end, some horses will remain; these will be given in order to the people with the highest fractional part of $m a_i$ until none is left. (In case of a tie, begin with the person with the lowest index.)

For example, consider $a_1 = 0.5, a_2 = 0.25, a_3 = 0.25$ ; that is, the first person contributes twice as much as each of the others. They manage to buy $m = 5$ horses. They are now distributed: the first person should get $m a_1 = 2.5$ horses, the second and third $m a_2 = m a_3 = 1.25$ . We first ignore the fractions, so the first person gets 2, and the other two 1 each. Now, the highest fractional part comes from the first person, so the first person gets an additional horse; that's 3 for the first person and 1 for the other two.

With $m = 6$ , the second person would get an additional horse (to 2), and with $m = 7$ , the third person would also get one more (to 2 as well).

The above example shows that increasing $m$ wouldn't make someone to get less horses. Is this true? That is, for any fixed $a_1, \ldots, a_n$ , is it true that if we raise $m$ , nobody will lose horses?