  We consider a special case of Dickson's lemma: for any two functions $f,g$ on
the natural numbers there are two numbers $i<j$ such that both $f$ and $g$
weakly increase on them, i.e., $f_i\le f_j$ and $g_i \le g_j$. By a
combinatorial argument (due to the first author) a simple bound for such $i,j$
is constructed. The combinatorics is based on the finite pigeon hole principle
and results in a descent lemma. From the descent lemma one can prove Dickson's
lemma, then guess what the bound might be, and verify it by an appropriate
proof. We also extract (via realizability) a bound from (a formalization of)
our proof of the descent lemma.
  Keywords: Dickson's lemma, finite pigeon hole principle, program extraction
from proofs, non-computational quantifiers.
