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To quote from my lecture notes:

When every subset of A has a lub and glb, we say that the order is a complete lattice, but this takes us beyond the syllabus. It is notable that $\mathbb{Q}$, ordered by $\leq$, is not a complete lattice but $\mathbb{R}$, ordered by $\leq$, is a complete lattice. This is the fundamental difference between $\mathbb{Q}$ and $\mathbb{R}$.

Please can someone explain why this is true? I can't see what the lub of $\mathbb{R}$ would be, in the same way I can't see a lub for $\mathbb{Q}$.

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    You are right. In the usual sense of lattice completeness, the reals under the ordinary order do not form a complete lattice.2012-12-18

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