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I have to proof the every finite lattice is also a complete lattice but I'dont known where to start.

From definition, a lattice is a poset in which every pair of elements has a least upper bound and a greatest lower bound. If the previous conditions are valid for any subset that the lattice is also complete.

How can I prove that every finite lattice is also complete? I need some hint.

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  • $\begingroup$ What would be the least upper bound of three elements x, y, and z? What about the least upper bound of n many elements? Now since your lattice is finite, you'll never have to find a bound on more than finitely many elements. Do you see where to go from here? $\endgroup$
    – HallaSurvivor
    Feb 6, 2020 at 20:42

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sup{ $a_1, a_2,.. a_n$ } = $a_1$ sup $a_2$ sup ... sup $a_n$

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  • $\begingroup$ This looks like a non-answer; can you please explain a bit more clearly what you are saying? $\endgroup$
    – kimchi lover
    Feb 7, 2020 at 19:27

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