There are topological and algebraic proofs.
Topological proof Let \sigma be the
dual cone of the given rational polyhedral cone. Let u_1, \dots, u_r be integral vectors so that \sigma = \{ x \mid \langle u_i, x \rangle \ge 0, 1 \le i \le r \}. Then the u_i's generate the dual cone \sigma^{\vee}; indeed, writing
C for the cone generated by u_i's, we have: \sigma \subset C^{\vee}, which must be the equality. Now, if
x is in the semigroup :S_\sigma = \sigma^\vee \cap \mathbb{Z}^d, then it can be written as :x = \sum_i n_i u_i + \sum_i r_i u_i, where n_i are nonnegative integers and 0 \le r_i \le 1. But since
x and the first sum on the right-hand side are integral, the second sum is a lattice point in a bounded region, and so there are only finitely many possibilities for the second sum (the topological reason). Hence, S_{\sigma} is finitely generated.
Algebraic proof The proof is based on a fact that a semigroup
S is finitely generated
if and only if its semigroup algebra \mathbb{C}[S] is a
finitely generated algebra over \mathbb{C}. To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the following statement: for any unital subsemigroup
S of \mathbb{Z}^d, : If
S is finitely generated, then S^+ = S \cap \{ x \mid \langle x, v \rangle \ge 0 \},
v an integral vector, is finitely generated. Put A = \mathbb{C}[S], which has a basis \chi^a, \, a \in S. It has \mathbb{Z}-grading given by :A_n = \operatorname{span} \{ \chi^a \mid a \in S, \langle a, v \rangle = n \}. By assumption,
A is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that \mathbb{C}[S^+] = \oplus_0^\infty A_n is a finitely generated algebra over A_0. Now, the semigroup S_0 = S \cap \{ x \mid \langle x, v \rangle = 0 \} is the image of
S under a linear projection, thus finitely generated and so A_0 = \mathbb{C}[S_0] is finitely generated. Hence, S^+ is finitely generated then.
Lemma: Let
A be a \mathbb{Z}-graded ring. If
A is a
Noetherian ring, then A^+ = \oplus_0^{\infty} A_n is a finitely generated A_0-algebra. Proof: Let
I be the ideal of
A generated by all homogeneous elements of
A of positive degree. Since
A is Noetherian,
I is actually generated by finitely many f_i's, homogeneous of positive degree. If
f is homogeneous of positive degree, then we can write f = \sum_i g_i f_i with g_i homogeneous. If
f has sufficiently large degree, then each g_i has degree positive and strictly less than that of
f. Also, each degree piece A_n is a finitely generated A_0-module. (Proof: Let N_i be an increasing chain of finitely generated submodules of A_n with union A_n. Then the chain of the ideals N_i A stabilizes in finite steps; so does the chain N_i = N_i A \cap A_n.) Thus, by induction on degree, we see A^+ is a finitely generated A_0-algebra. == Applications ==