Suppose \(\{ a_n \}\) is a bounded increasing sequence. Let \(L\) be the least upper bound of the set \(\{a_n | n \ge 1\}\). In other words, for every \(ε \gt 0\), there is some integer \(N\) such that:
\[ a_N \gt L - ε \]
If \(\{ a_n \}\) is increasing, then \(a_n \ge a_N\) for every \(n \gt N\). Thus, if \(n \gt N\), we have:
\[ a_n \gt L - ε \]
So:
\[ 0 \le L - a_n \lt ε \]
Since \(a_n \le L\), then whenever \(n \gt N\):
\[ | L - a_n | \lt ε \]
So \(\lim_{n \to \infty} a_n = L\) (by definition of a limit of a sequence). A similar proof (using the greatest lower bound) works if \(\{a_n\}\) is decreasing.