Page 170 - 'Blast_Into_Math

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

Let’s think about X − . Since > 0, we know that

X − < X.

By the definition of least upper bound, X − is not an upper bound for the sequence. This means that
there is some N ∈ N such that x N >X − . Now, let’s think about where the sequence is going. It

is increasing, which means that x N+1 ≥ x N , and in fact

x N ≤ x n , ∀n> N.

Putting these facts together, we have

X − < x N ≤ x n ≤ X< X + , ∀n> N,

so

|x n − X| < ∀n> N.

Exercise: Use the same strategy to prove that a decreasing sequence converges if and only if it is bounded

below, in which case the sequence converges to its greatest lower bound.

♥

Remark 7.2.6 Almost all mathematics is achieved through mathematical teamwork. Remember to take
pride in your contribution to the team!

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