True here is the proof if s n a and s n t n b thent n

3. True. Here is the proof: If s
n
→ A and s
n
+ t
n
→ B , then t
n
= ( s
n
+ t
n
) − s
n
→
B − A .
4. Not true. A counterexample: s
n
= 0, and s
n
t
n
= 0 · n = 0 converges, but t
n
= n
diverges. �
• 4.2-9* Prove Theorem 4.2.12: Suppose that ( s
n
) and ( t
n
) are sequences such
that s
n
≤ tn for all n ∈ N .
1. If lim s
n
= + ∞ , then t
n
= + ∞ .
2. If lim t
n
= −∞ , then lim s
n
= −∞ .
Proof:
• 4.2-12* Suppose that ( s
n
) converges to s . Prove that ( s
2
n
) converges to s
2
directly without using the product formula of Theorem 4.2.1(c).
18

Since ( s
n
) converges, it is bounded: | s
n
| ≤ M for all n ; and for the � , there is N > 0
such that
| s
n
− s | <
�
| s | + M
, ∀ n > M.
Then
| s
2
n
− s
2
| = | s + s
n
|| s − s
n
| ≤ ( | s | + | s
n
| ) | s
n
− s | ≤ ( | s | + M ) | s
n
− s |
< ( | s | + M ) ·
�
| s | + M
= � , ∀ n > N. �
• 4.2-14* Prove lim
�
1
n
−
1
n +1
�
= 0 .
Proof: To show: For any � > 0, there exists N > 0 such that
�
�
�
�
1
n
−
1
n + 1
�
�
�
�
< � , ∀ n > N.
Consider |
1
n
−
1
n +1
− 0 | =
1
n
2
+ n
<
1
n 2
<
1
n
. For the � , we take N >
1
�
so that the desired
inequality holds for any n > N . �