Monotone Convergence Theorem (Real Analysis)/Increasing Sequence

Theorem
Let $\left \langle {x_n} \right \rangle$ be increasing and bounded above.

Then $\left \langle {x_n} \right \rangle$ converges to its supremum.

Proof
Suppose $\left \langle {x_n} \right \rangle$ is increasing and bounded above.

By the Continuum Property, it has a supremum, $B$.

We need to show that $x_n \to B$ as $n \to \infty$.

Let $\epsilon > 0$.

By the definition of supremum, $B - \epsilon$ is not an upper bound.

Thus:
 * $\exists x_N: x_N > B - \epsilon$

But $\left \langle {x_n} \right \rangle$ is increasing.

Hence:
 * $\forall n > N: x_n \ge x_N > B - \epsilon$

But $B$ is still an upper bound for $\left \langle {x_n} \right \rangle$.

Then:

Hence the result.