Recall that given a representation $\pi$ of $\mathfrak{sl}_n$, a weight $\mu$ is said to be of highest weight if its corresponding weight vector is annihilated by all the positive root spaces (1). Now suppose I give another definition for what it means for a weight to be of highest weight:
(2) Given weights $\mu_1 = a_1L_1 +\ldots a_nL_n$ and $\mu_2 = b_1L_1 + \ldots b_nL_n$ of some representation of $\mathfrak{sl}_n$, we say that $\mu_1$ is higher than $\mu_2$ (denoted $\mu_1 > \mu_2$) if the first $i$ for which $a_i - b_i$ is non-zero (if any) is positive. A weight $\mu$ is then said to be of highest weight if for any other weight $\nu$, $\nu \leq \mu$.
Now I can see that not (1) implies not (2), but how can I see that (1) implies (2) to show that the definitions are equivalent? I can show this in the case that $\pi$ is an irreducible representation and hence a highest weight representation, for then the weights of such a representation have a very nice description. They are all of the form
$\text{(highest weight) minus (some positive roots)} $
and I can see why (1) will imply (2) in this case. However in the more general case that $\pi$ is not irreducible, how will (1) imply (2)?
Thanks.