Suppose that I have $n$ intervals $[a_1,b_1],...,[a_n,b_n]$. Then I know that the Cartesian Product $[a_1,b_1] \times ... \times [a_n,b_n]$ is a complete lattice with the meet and join operations $x \vee y = ( \max(x_1,y_1),...,\max(x_n,y_n)), x \wedge y = (\min(x_1,y_1),...,\min(x_n,y_n)).$
Now suppose that I have intervals $[a_i,b_i]$ where $i \in \mathcal{I}$ is an index in an uncountable set. Is the infinite product $\Pi_{i \in \mathcal{I}} [a_i,b_i]$ a complete lattice under the operations
$(x \vee y)_i = \max(x_i,y_i)$ $(x \wedge y)_i = \min(x_i,y_i)$?
Yes. $x$ and $y$ are definitely less than or equal to their supremum.
Also, for any $z$ $$x \vee y \le z \iff (\forall i, max(x_i, y_i)\le z_i) \iff (\forall i, x_i \le z_i \text{ and } y_i \le z_i) \iff z\le x\text{ and } z\le y$$
So it does meet both defining properties of the supremum.