Questions to redo

Question 1

Definition of subsequence

Answer 1

A subsequence of a sequence (xn)∞_n=1
is a sequence of the form (xkn)∞_n=1
for some strictly increasing natural numbers k1 < k2 < k3 < . . . .

Question 2

Bolzano-Weierstraß theorem

Answer 2

Every bounded real sequence has a convergent subsequence.

Question 3

Intermediate Value Theorem

Answer 3

Let f : [a, b] → R be a continuous function and assume that d is a number such that f(a) < d < f(b) or f(b) < d < f(a). Then there exists a point c ∈ (a, b) such that
f(c) = d.

Question 4

Extreme value theorem

Answer 4

∃u, v ∈ [a, b] such that ∀x ∈ [a, b], f(u) <= f(x) <= f(v).