Page 190 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

But this is a contradiction because we can re-arrange to

1
x 1 x 2 x n
x − + + ... + = ,
10 10 2 10 n 10 n

but by the induction hypothesis

1
x 1 x 2 x n
0 ≤ x − + + ... + < .
10 10 2 10 n 10 n

So if x n+1 =9, then

1 10 x n+1 +1
x 1 x 2 x n
x − + + ... + < = = ,
10 10 2 10 n 10 n 10 n+1 10 n+1
which subtracting x n+1
n+1 from both sides means
10

1
x 1 x 2 x n x n+1
x − + + ... + + < .
10 10 2 10 n 10 n+1 10 n+1
So, x n+1 is the (unique) largest element of S , and

1
x 1 x 2 x n x n+1
0 ≤ x − + + ... + + < .
10 10 2 10 n 10 n+1 10 n+1

We have also proven that the digits x m for m> n +1 cannot all be 9. This sets the induction escalator
into motion. To complete the proof of the Proposition, by the Geometric Sequence Lemma, the sequence

1 1 n
=
10 n 10

converges to 0. This means that for any > 0, there exists M ∈ N such that for all N> M,

1
x 1 x 2 x N
0 ≤ x − + + ... + < <.
10 10 2 10 N 10 N

We can re-arrange this to

N
x n

− x <, ∀N> M.

10
n
n=1

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