I if s n is an unbounded sequence then either s n or

(i) If ( s
n
) is an unbounded sequence, then either s
n
→ + ∞ or s
n
→ -∞ .
(j) If ( s
n
) is a bounded sequence and α = sup { s
n
} , then ( s
n
) has a
subsequence which converges to α .
(k) For any given sequence ( s
n
), ( s
n
) has a convergent subsequence ( s
n
k
).
4

(l) For any bounded sequence ( s
n
), lim sup s
n
= sup { s
n
} .
(m) It α is a subsequential limit of a bounded sequence ( s
n
), then α ≤
lim sup s
n
(n) If every subsequence of a sequence ( s
n
) is convergent, then ( s
n
) itself
must be convergent.
5

(o) If ( s
n
) is a divergent sequence, then some subsequence of ( s
n
) must
diverge.
(p) If ( s
n
) is unbounded above, then ( s
n
) has an increasing subsequence
( s
n
k
) which diverges to + ∞ .
6

2. Let ( s
n
) be a positive sequence such that
lim
n →∞
s
n +1
s
n
= L > 1 .
Prove that s
n
→ + ∞ . Hint: lim
n →∞
x
n
= + ∞ for any number x such that
x > 1.
7

3. Limits: Determine the convergence or divergence of each of the following
sequences. If the limit exists, state what it is. (Hint: Use the previous problem,
and Theorem 3 from Section 17.)
(a) s
n
=
n 3
3 n
(b) s
n
=
n !
n 10
(c) s
n
=
n n
n !
8

4. For each of the following sequences, determine: (1) Whether the sequence con-
verges; if it does, give the limit. (2) If the sequence does not converge, find the
set of subsequential limits (if any) and give lim sup and lim inf.
(a) s
n
=







1
n + 1
, n odd
3
2 n - 1
, n even
(b) s
n
=













1
n
, n = 3 , 6 , 9 , · · ·
n
n + 1
, n = 1 , 4 , 7 , 10 , · · ·
n 2 , n = 2 , 5 , 8 , 11 , · · ·
(c) s
k
= k sin
2
kπ
4
(d) s
n
=
1
2 n
+ cos
nπ
6
9