(In a way this is a rephrasing of Iosif's answer, but from a different perspective. My main intention is to point out the connection to matroids.)

The inequality follows from:

**Fact.** The polytope $K = \Bigl\{\alpha \in [0,1]^n: \sum_{i = 1}^n\alpha_i = N\Bigr\}$ is the convex hull of indicator vectors of subsets of $[n] = \{1, \ldots, n\}$ of size $N$.

Assuming this fact, and because any linear function over a polytope achieves its minimum at a vertex, we have that $\sum_{i = 1}^n{\alpha_i x_i} \ge \sum_{i \in S} x_i$ for some set $S$ of size $N$. The right hand side is then clearly minimized for $S = \{1, \ldots, N\}$ if $x_1 \le x_2 \le \ldots \le x_n$.

To show the Fact, we need to show that every extreme point $\alpha$ of $K$ is an indicator vector of a set of size $N$. This is clearly true if $\alpha \in \{0,1\}^n$, so assume, towards contradiction, that for some $i$ we have $0 < \alpha_i < 1$. Then there must be at least one other $j \neq i$ such that $0 < \alpha_j < 1$, and we can write $\alpha$ as a convex combination of two vectors in $K$, one in which we add a tiny $h$ to $\alpha_i$ and subtract $h$ from $\alpha_j$, and one in which we reverse the signs. This means that $\alpha$ is not an extreme point, a contradiction. You can see that this argument is essentially the same as Iosif's.

Another way to state the Fact is that $K$ is the base polytope of the uniform matroid over $[n]$ of rank $N$. A vast generalization is the characterization of the facets of the base polytope of any matroid, due to Edmonds: see, e.g., Section 10.7 in these notes.