Example of a bounded lattice that is NOT complete I know that every complete lattice is bounded. Is there a simple example for a bounded lattice that is not complete?
Thank you 
 A: 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.
A: Clearly, $\mathbb{Q}\cap[0,1]$ is a bounded lattice. Pick an irational number $x\in[0,1]$ and observe that $\mathbb{Q}\cap [0,x)$ has no join.
A: Update: My answer below is wrong! (Thanks to bof for pointing that out.) I will leave the answer here because I think my mistake and bof's comment could maybe be instructive.
Let $\mathbb{N}$ be the set of natural numbers. Let $\mathcal{P}_{fin}(\mathbb{N})$ denote the collection of finite subsets of $\mathbb{N}$. Then $L=\mathcal{P}_{fin}(\mathbb{N})\cup\{\mathbb{N}\}$ is a bounded lattice under inclusion. However, it is not complete since $\{0\}\cup\{2\}\cup\{4\}\cup\{6\}\cup\ldots$ is not in $L$.
