Originally Posted by

**Sudharaka** Hi wopashui,

This problem is incorrect. There are sets for which there exist lower bounds but not a greatest lower bound. For example let,

$\displaystyle \mathbb{Q}$ be the set of Rational numbers and $\displaystyle A=\left\{a\in\mathbb{Q}~|~a\in\left(\sqrt{2},5 \right) \right\}.$ Clearly there are lower bounds for the set A, such as, 1,0,-1 etc. But a greatest lower bound does not exist. Since if you take any lower bound (say $\displaystyle l\in\mathbb{Q}$) there exist infinity many rational elements in between $\displaystyle l$ and $\displaystyle \sqrt{2}$.