sequences and dilworth's theorem

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Can someone please help me understand the last inequality. Specifically how does it follow that m-1 + 1/n-1 >= m.

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In fact, $$m - 1 + \frac{1}{n - 1} \gt m - 1.$$ Therefore, size of antichain $\gt m - 1$; since size of antichain is an integer, this inequality implies size of antichain $\ge m$.