For large n s n s and s n 1 s therefore s 1 4 2s 7 4

4

6. (a) Suppose ( a
n
) is increasing. Then a
n +1
≥ a
n
for all n . If k ≥ 0, then
ka
n +1
≥ ka
n
for all n ; if k < 0, then ka
n +1
≤ ka
n
for all n . In either
case, ( ka
n
) is monotone.
(b) Counterexample: Let ( a
n
) = (1 , 2 , 2 , 3 , 3 , 4 , 4 , . . . ) and let ( b
n
) = (1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , . . . ).
( a
n
) and ( b
n
) are increasing sequences. The sequence a
n
/b
n
is:
parenleftbigg
1 , 2 , 1 ,
3
2
, 1 ,
4
3
, . . .
parenrightbigg
which is not monotone.
7. It is clear that s
n
is increasing.
Claim s
n
< 3 for all n . By induction: Let S be the set of positive integers
for which s
n
< 3.
1 ∈ S : s
1
=
√
6 < 3. Thus 1 ∈ S .
Assume that k ∈ S . That is, assume s
k
< 3.
Prove that k + 1 ∈ S . That is, prove
5