Completeness (Intuition) Math Example 1

Solution

1. 1
   Consider the decimal approximations of $\sqrt{2}$: $1, 1.4, 1.41, 1.414, 1.4142, \ldots$
2. 2
   Each term is rational (a terminating decimal). The sequence converges — each term is closer to $\sqrt{2}$ than the last.
3. 3
   But $\sqrt{2} \notin \mathbb{Q}$. In $\mathbb{Q}$, this sequence has no limit — the limit 'falls through a gap.'
4. 4
   In $\mathbb{R}$, $\sqrt{2}$ exists, so the limit exists. $\mathbb{R}$ is complete; $\mathbb{Q}$ is not.