A common problem form.

Recently I've encounted a few problems, and they all have the same keyword: $infinite$ . For example, this one:

A positive integer is considered 'good' when it can be expressed as the sum of $2$ perfect squares, e.g.:

$8 = 2^{2}+2^{2} , 9=3^{2}+0^{2} , 10=3^{2}+1^{2}$

Prove that there exists infinitely many triples of consecutive positive integers, all of which are good.

Exercises:

1) Solve the problem.

2)What other methods do you use when encountering proof problems that contain $'infinite'$ ?

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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}`	$\boxed{123}$