I am reading the paper Permutations and the 15-Puzzle by Peter Trapa (2004). Section 2. Parity of integers says the following:
Let's start by considering the integers $\mathbb{Z}$. One of the interesting features of $\mathbb{Z}$ is the existence of the notion of parity, that is evenness and oddness. Somewhere in our distant past we learned the basic rules $$\text{even plus even equals even}$$ $$\text{even plus odd equals odd}$$ $$\text{odd plus odd equals even,}$$ and from those we can quickly derive the analogous rules for multiplication, as well as slightly more complicated rules such as $$\text{the sum of an odd number of odd numbers is odd,} \tag{*}$$ and so on.
This is so obvious and natural that we hardly ever think about it. But the notion of parity can be surprisingly useful. Consider the following$^2$. Suppose there are 33 people at a party. Then we claim that it follows that at least one person at the party knows an even number of people. (Here we assume that acquaintance is mutual — if you know someone, then they know you — and we also allow for the possibility of total strangers attending who don’t know anyone at all. In the latter case, the problem is trivial since zero is an even number.) Here is how to prove our claim. For convenience, label the people at the party as $1, 2, \dots, 31$. Let $n_i$ denote the number of acquaintances that the $i$th person has. Since acquaintance is mutual, the sum $n_1 + n_2 + \dots + n_{31}$ must equal twice the total number of pairs of acquaintances. Thus the sum $n_1 + n_2 + \dots + n_{31}$ is even. By (*) above, at least one of the $n_i$'s must be even, and the claim is proved.
It is this part that I am unsure about:
By (*) above, at least one of the $n_i$'s must be even, and the claim is proved.
We have (1) that (*) says that the sum of an odd number of odd numbers is odd, and (2) that the sum $n_1 + n_2 + \dots + n_{31}$ is the sum of an odd number of numbers, the sum total of which is even. It seems to me that the author's reasoning is implying that the facts (1) and (2) therefore imply that at least one of the $n_i$s must be even. However, I don't see how this is valid reasoning. Rather, it seems to me that, for such a proof, you would first need to prove that, if the sum of an odd number of numbers equals an even number, then at least one of the numbers (that is being summed) must be even; but the author did not prove this, and instead jumped straight to the conclusion.
Am I correct here? If not, then what am I missing?