Im reading Chapter12 of Carothers' Real Analysis, 1ed talking about the Stone-Weierstrass theorem. Here is a definition of Lattice,
See. every pair of elements has both a sup and inf in the lattice. So it seems to tell us that lattice is always complete??? What happened? If not, can you show me a incomplete lattice?
The definition of a lattice only requires each pair of elements to have a sup and an inf, but from this you can deduce that each triple of elements must also have a sup and an inf. If you continue the argument then you find that in a lattice $L$ each non-empty finite subset $X\subseteq L$ actually has a sup $\mathop\vee X\in L$ and an inf $\mathop\wedge X\in L$ satisfying $$ \text{$f\leq h$ for all $f\in X$}\quad\Leftrightarrow\quad\mathop\vee X\leq h $$ and $$ \text{$h\leq f$ for all $f\in X$}\quad\Leftrightarrow\quad h\leq\mathop\wedge X $$ for all $h\in L$.
Although it looks very similar, this is not the same as $L$ being complete; the key point is that we can only deduce that each (non-empty) finite subset of $L$ has a sup and an inf. For $L$ to be complete we would need every subset of $L$ to have a sup and an inf.
See this answer to a similar question for a more detailed discussion and an example of an incomplete lattice.