Suppose that p is a prime number and k > 0.
There are pk-1
multiples of p in {1, ..., pk}.
Hence pk-pk-1 numbers
in this set are prime to p.
But then the same number is prime to pk.
In particular, 
(pk) =
pk-pk-1 = (p-1)pk-1,
proving the lemma.
Suppose that p is a prime number and k > 0.
There are pk-1
multiples of p in {1, ..., pk}.
Hence pk-pk-1 numbers
in this set are prime to p.
But then the same number is prime to pk.
In particular, (pk) =
pk-pk-1 = (p-1)pk-1,
proving the lemma.