N n ℕ s n t n 1 but tn diverges to 9 suppose s n t

= n, ∀ n ∈ℕ . ( S
n
* T
n
)  1, but T
n
diverges to +
9) Suppose S
n
≤ T
n
, ∀ n ∈ℕ and lim(S
n
) = + ∞ . Given M ∈ ℝ , there exist n ≥ N implies that S
n
 for n ≥ N, T
n
≥ S
n
>M  lim(T
n
) = +
Suppose S
n
≤ T
n
, ∀ n ∈ℕ and lim(T
n
) = - ∞ . Given M ∈ ℝ , there exist n ≥ N implies that T
n
 for n ≥ N, T
n
≤ S
n
< M  lim(S
n
) = -
12) Suppose S
n
is convergent, then there exist M
1
∈ ℝ such that |S
n
|<M
1
, ∀ n ∈ℕ
Let M= | M
1
+|s||. Given 𝜀 >0, there exist N such that n>N  | S
n
- S |<
𝑀
* S
n
≤ S
n
2
|S
n
2
- S
2
|= |S
n
- S||S
n
+ S|< | M
1
+|s||| S
n
- S |< 𝜀
| S
n
- S |<
𝑀
(M)= 𝜀
 S
n
2 approaches S 2
∞
>M
∞
<M
∞
.
𝜀
𝜀

18) Suppose a ≤ S
n
∀ n ∈ℕ , then a ≤ lim(S
n
)=s
-Suppose s<a. Then ∀ 𝜀 >0 , there exist N ∈ℕ such that n ≥ N  | S
n
- S |<
Section 4.3
k
k