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 $\mathbb{Q}$.