Go Up | 
Next Problems | 
Prev. Problems |
Glossary |
Map
Number Sequences: Problems
1. If a = {
} and b =
{
} are two sequence, we can define their sum,
difference, product, and quotient as
- {}+ {} =
{+ }
- {}- {} =
{- }
- {}* {} =
{* }
- {} / {} =
{ / } (if is never zero)
Prove or disprove the following statements:
- if a and b are both increasing, so is their sum
- if a and b are both increasing, so is their difference
- if a and b are both increasing, so is their product
- if a and b are both increasing, so is their quotient
- if a and b are both bounded, so is their difference
- if a and b are both bounded, so is their quotient
2. Let a = {} and b =
{} be two sequence. Prove or disprove the following
statements:
- if a is convergent and b is divergent, then their sum is divergent
- if a and b are divergent, then their sum is divergent
- if a + b and a - b are both convergent, then
a and b are convergent
- if a - b and a + b are both divergent, then
a and b are divergent
- if a is convergent and b is divergent, then a *
b is divergent
3. If {} is increasing, then the sequence
{} with = 
is also increasing.
4. Let {
} be the sequence of open
intervals = (-1/n, 1/n). Show that the sequence is
decreasing in the sense that 
. Show that there is one
and only one number that belongs to for any n.
5. Show that if a sequence {} converges to a
negative limit, then its terms eventually become and remain negative.
6. Criticize the following argument: every neighborhood of 1 contains all terms
of the sequence j}.gif)
except perhaps the
term -1. Therefore, every neighborhood of 1 contains all except a finite number of terms
of the sequence, so that the sequence converges to 1.
7. Criticize the following argument:
sin(n) / n = ( sin(n) ) * 1/n) = sin(n) ) * 0 = 0
Find (with prove) the actual limit.
8. Show that if the sequence {} is
convergent, then the sequence {} with = converges
to the same limit.
9. Give examples of sequence {} and
{}, both of which diverge to positive infinity, for
which:
- their quotient converges to infinity
- their quotient converges to zero
- their quotient converges to 2
10. Let {} be defined recursively as
follows
Show that {} converges and find its limit
11. Prove or disprove the following statements:
- every sequence has an increasing subsequence
- every sequence has a bounded subsequence
- every unbounded sequence has a subsequence that diverges to positive or negative
infinity
- if a sequence has a greatest term, then every subsequence of that sequence has a
greatest term
12. Let 
and 
be two sequences. Prove that
- lim sup (. + )
lim sup () + lim sup
()
How are the lim inf's related ? Find an example where strict inequality holds.
13. Let be a sequence
of positive numbers. What is the relation between the lim inf and the lim sup of { } and the lim inf and the lim sup of { 1 / } ?
14. Let be a sequence
of real numbers. What is the relation between the lim inf and the lim sup of { } and the lim inf and the lim sup of { - } ?
15. If a and b are any real numbers with a < b, then give an example of a
sequence whose lim sup is equal to b and whose lim inf is equal to a.
16. Find a single sequence such that for any real number c you can extract a
subsequence that converges to c.
17. Give another proof of the Bolzano-Weierstrass theorem as follows: If
is a bounded sequence, define
= inf{ 
, 
, 
, ... }. Then each
is finite, and the sequence {} is increasing and bounded above. Therefore, by our theorem on
bounded, increasing sequences, this new sequence must have a limit.
Note: The proof is not done. You need to construct a convergent subsequence
using the above 'hint'.
18. Give another proof of the theorem that every increasing (decreasing)
sequence that is bounded above (below) converges as follows: If a sequence is increasing
and bounded above, then it must be bounded. Therefore, by the Bolzano-Weierstrass
theorem, it has a convergent subsequence. Then it follows that the entire sequence
converges.
Note: The proof is not done. You need to provide a careful justifications of the
'hints' given above.
19. Prove that a sequence converges if and only if the lim sup and the lim inf of
the sequence are equal.
20. Find the supremum and the infimum of the empty set.
Go Up | Next Problems | Prev. Problems |
Glossary |
Map
= 1
for n
> 1