Convergence and the Limit of Complex Sequences

(4) Suppose r1 is convergent. Let given p. Suppose 0 < p. Then there exists n such that for all natural numbers m, l such that n ≤ m and n ≤ l holds |r1(m)−r1(l)| < p. (5) If for every n holds r1(n) ≤ p, then for all natural numbers n, l holds ( ∑κ α=0(r1)(α))κ∈N(n+ l)− ( ∑κ α=0(r1)(α))κ∈N(n) ≤ p · l. (6) If for every n holds r1(n) ≤ p, then for every n holds ( ∑κ α=0(r1)(α))κ∈N(n) ≤ p · (n+ 1). (7) If for every n such that n ≤ m holds r2(n) ≤ p·r3(n), then ( ∑κ α=0(r2)(α))κ∈N(m) ≤ p · ( ∑κ α=0(r3)(α))κ∈N(m). (8) Suppose that for every n such that n ≤ m holds r2(n) ≤ p · r3(n). Let given n. Suppose n ≤ m. Let l be a natural number. If n+ l ≤ m, then ( ∑κ α=0(r2)(α))κ∈N(n+ l)− ( ∑κ α=0(r2)(α))κ∈N(n) ≤ p · (( ∑κ α=0(r3)(α))κ∈N(n+ l)− ( ∑κ α=0(r3)(α))κ∈N(n)).