Prove usual $\leq$ on R in [0,1] is a lattice.

Question

Let $R$ be the set of real numbers in $[0,1]$ and $\leq$ be the usual operation of "less than or equal to" on $R$. Show that $(R,\leq)$ is a lattice.

Definition of Lattice: A poset in which every pair of elements has a GLB and LUB.

Answer 1

I’ll use $\land$ for the lattice operation of meet, infimum, or greatest lower bound and $\lor$ for the operation of join, supremum, or least upper bound. (All of these terms are used.)

Suppose that $x,y\in\Bbb R$; when is a real number $a$ a lower bound for the set $\{x,y\}$? Clearly it’s when $a\le x$ and $a\le y$, so $a$ is a lower bound for $\{x,y\}$ if and only if $a\le\min\{x,y\}$. Let $L(x,y)=L(y,x)$ be the set of all lower bounds of $\{x,y\}$; then we’ve just seen that

$$L=\{a\in\Bbb R:a\le\min\{x,y\}\}\;.$$

Does $L$ have a largest element? Yes: $\max L=\min\{x,y\}$. In other words, $\min\{x,y\}$ is the greatest lower bound, or meet, of $x$ and $y$, and we define $$x\land y=\min\{x,y\}\;.$$

With this as a model can you do the other half of the argument? It’s very similar.