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Latest Proof: Pascal's Wager on 4 July 2009 by Matt Westwood

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Five Color Theorem [1]--Joe 00:57, 11 August 2008 (UTC)
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Proof of the Week

Monotone Convergence Theorem

Theorem

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\mathbb{R}$ .

Increasing Sequence

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

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

Decreasing Sequence

Let $\left \langle {x_n} \right \rangle$ be decreasing and bounded below.

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

Proof

Proof for Increasing Sequence

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

Let its supremum be $B$ .

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

Let $ε > 0$ .

Since $B - ε$ is not an upper bound, by the definition of supremum.

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 $\left \langle {x_n} \right \rangle$ .

	$\forall n > N: B - \epsilon$	$<$	$x_n \le B$
$\Longrightarrow$	$\forall n > N: B - \epsilon$	$<$	$x n < B + ε$	Real Plus Epsilon
$\Longrightarrow$	$\forall n > N: \left\|{x_n - B}\right\|$	$<$	$ε$	Negative of Absolute Value: Corollary