TITLE: Example of a bounded lattice that is NOT complete
QUESTION [7 upvotes]: I know that every complete lattice is bounded. Is there a simple example for a bounded lattice that is not complete?
Thank you

REPLY [6 votes]: I guess the simplest example is
$$\{\frac1n:n\in\mathbb N\}\cup\{-\frac1n:n\in\mathbb N\}$$
with the usual ordering of real numbers. Perhaps the most natural example is
$$\{X\subseteq\mathbb N:X\text{ is finite or }\mathbb N\setminus X\text{ is finite }\}$$
ordered by set inclusion. In the first example the set of all negative numbers, in the second example the set of all finite sets of even numbers has no least upper bound.