Notes on Chapter 1

Exercises

Ex 3.9

Let \(b_0 ≔ K\) and take some \(a \in A\) to be \(a_0 \coloneqq a\).

Then \(\forall n \in \mathbb{N}_0\) let \(c_{n} = \frac{a_n + b_n}{n}\)

Case 1 \(c_n\) is an upper bound of A.

If \(c_{n} \geq a \quad \forall a \in A\) then

\[ \left[ a_n, c_n \right] \cap A = \left[ a_n, b_n \right] \cap A \neq \emptyset \]

Let \(\left[ a_{n+1}, b_{n+1} \right] = \left[ a_n, c_n \right]\)

Case 2 \(c_n\) is not an upper bound of A.

If there exists some \(a\)