6 More about the real numbers

Question 1

Theorem 6.1:
Nested interval theorem

Answer 1

Let (an)n∈N and (bn)n∈N be two sequences of real numbers such that an ≤ bn and [a(n+1), b(n+1)] ⊆ [an, bn] for every n ∈ N.
Then ∩n∈N [an, bn] ≠ ∅.

∩n∈N [an, bn] = {x∈R : x∈[an,bn] for all n∈N}.

i.e.
* The intersection of all the intervals is not empty
* There exists at least one point that is in all the intervals [an, bn]

Question 2

Corollary 6.1:
[0,1]

Answer 2

No real sequence contains every element of [0,1].

Question 3

Definition 6.1:
Finite and countable set

Answer 3

• Set S is finite if there exists a bijection between S and {1, … , n} for some n ∈ N (n = cardinality of S).
• Set S is countable if there exists an injective map from S to N.

Question 4

Theorem 6.2 / Corollary 6.2:
Countability in relation to:
1. the interval [0,1]
2. the set R.

Answer 4

1. The interval [0,1] is uncountable.
2. The set R is uncountable.