Mathematics for Economics
Fall 2021
HOMEWORK-1
1.
(a). Explain the difference between a sequence and a set.
(b). What is wrong with the exercise 12.2 on page 259.
2. Consider the following sequence
1, 2, 3, 1, 2, 3, 1, ... . . . .
(a). Write down two sequences that are subsequences of this sequence.
(b). Is there a sequence whose all terms are in the set {1, 2, 3} that is not a
subsequence of this sequence? Prove.
3. Give the definition of a bounded sequence in Rn . Show that a convergence
sequence in Rn is bounded.
4. Find all accumulation points of the sequence
1
1
1
1, 3, 5, , 7, 9, 11, , 13, 15, 17, , . . .
2
4
8
5. Find the limit
√
k
k
lim ( √
,
).
k→∞
k−1 k−5
6. Find the interior of the set
{(x, y) ∈ R2 : x − y ≥ 0, x ≥ 0} ∪ {(x, y) ∈ R2 : x − y < 0, x < 0}.
7. Find the boundary of the set
{(x, y) ∈ R2 : |x| + |y| ≤ 1 and y ≤ 0} ∪ {(x, y) ∈ R2 : |x| + |y| < 1 and y > 0}.
8. Solve problems 12.29 and 12.30 on page 272.