Well done!

When you have a multi-part proof it helps (at least the reader…) to introduce each part with a kind of announcement so it's clear what's to come next:

We will first show that [blah blah]

the proof, part one

.

.

Now we can prove that [blah blah]

following part of the proof

.

.

&c.

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A little problem which could interest you in the context.

The period of the decimal development of 1/p, p a prime other than 2 or 5 (which divide the base 10), has of course a length at most p-1. But some reach that maximal length, and others not:

1/3 = 0.**3**333… has period length 1 < p-1 = 3-1 = 2

1/7 = 0.**142857**142857… has maximal period length p-1 = 6

1/11 = 0.0**90**90… has period length 2

1/13 = 0.0**769230**769230… has period length 6

For what primes is the period length maximal? What about other bases, 2 for example?