Monotone Convergence Theorem (Real Analysis)/Increasing Sequence

Theorem
Let $\sequence {x_n}$ be an increasing real sequence which is bounded above.

Then $\sequence {x_n}$ converges to its supremum.

Proof
Suppose $\sequence {x_n}$ 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 $\epsilon \in \R_{>0}$.

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

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

But $\sequence {x_n}$ is increasing.

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

But $B$ is still an upper bound for $\sequence {x_n}$.

Then:

Hence the result.