Definitions

Question 1

Cauchy

Answer 1

∀ε > 0, ∃n0 ∈ N s.t. ∀(n, m ∈ N with n, m ≥ n0), |xn − xm| < ε.

Question 2

Leibniz test

Answer 2

eventually positive, eventually decreasing and limit 0

Question 3

Subsequence

Answer 3

A subsequence of (xn): is a sequence (xkn): where kn∈ N and kn<kn=1

Question 4

Bolzano-Weierstrass theorem

Answer 4

Every bounded real sequence has a convergent subsequence