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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< f_j and g_i < 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 byarXiv:1503.03325v1 fatcat:vrmlu54jsjfyzkxy57zk74umcy