Overcomplicating things

This is a proof - based question.

$a(n) = (n^2 + n + 1)^2 + 1$ $b(n) = \frac{a(1)}{a(2)} + \frac{a(1)a(3)}{a(2)a(4)} + \cdots + \frac{a(1)a(3) \cdots a(2n - 5)a(2n - 3)}{a(2)a(4) \cdots a(2n - 4)a(2n - 2)} + \frac{a(1)a(3) \cdots a(2n - 3)a(2n - 1)}{a(2)a(4) \cdots a(2n - 2)a(2n)}$ Prove that $b(n) < \frac{1}{2}$ .

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$