Page 161 - 'Blast_Into_Math

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

2

You have proven in the exercises that

√
2 /∈ Q,

which shows that the set

2
S = {x ∈ Q such that x < 2},

is bounded above, but its least upper bound is not in Q. .

The sequence

1 √
x n = − + 2
n

is increasing because

1 1
< ∀n ∈ N
n +1 n

which means that

1 √ 1 √
x n+1 = − + 2 > − + 2= x n .
n +1 n

1
We have proven that the sequence converges to 0, so by the LAMP, the sequence
n
1
t n = − , lim t n = −0= 0,
n n→∞

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