经济学问题集代写 Economics 426代写 经济学代写 经济学代做

Economics 426: Problem Set 5

经济学问题集代写 I. Let {xn} be a sequence in Rn . Show that if xn is convergent, then the sequence must be bounded. II. Let A := (0, 1) × {0} be a subset of R2.

Points Possible: 100

Due Date: Wednesday, April 1, 2:00pm (online).

I.

Let {x_n} be a sequence in Rⁿ . Show that if x_n is convergent, then the sequence must be bounded.

II. Let A := (0, 1) × {0} be a subset of R².

A. Identify A as open, closed, both or neither. Explain.

B. Identify the closure, interior and boundary of A. Explain.

C. Is A compact? Explain. If not, provide an open cover that does not admit a finite subcover.

D. Is A connected? Explain. If not, write A as the union of two non-empty, disjoint, closed (relative to A) sets.

III. 经济学问题集代写

Recall that we define set summation in R^N as follows: Let A ⊂ R^N , B ⊂ R^N . Then A + B := {x | x = a + b, a ∈ A, b ∈ B}. Let A, B ⊂ R² where

A := {(x, y) | 1 ≤ x ≤ 2, y = −x}; and

B := {(x, y) | 0 ≤ x ≤ 1, 1 ≤ y ≤ 3}.

Described mathematically A + B. That is, fill in the blank — A + B := {(x, y)| . . .}. Sketch A, B and A + B.

B. Is x_n convergent? Explain.  

C. If your answer to part B is “yes,” find the limit. Explain.

D. If your answer to part A is “yes” and to part B “no,” then find a convergent subsequence and its limit. Demonstrate convergence.

V. 经济学问题集代写

Consider each of the following functions of R to R. In each case state whether the function is continuous at 0. Explain. If the function is not continuous at 0, find an open set in R such that the pre-image under the function is not open.

A. f(x) = 3x + 2  

B. g(x) = −x for x ≤ 0 and g(x) = 0.001 for x > 0.

C. h(x) = |x| (absolute value).

VI. 经济学问题集代写

Let S ⊂ R² , S compact (closed and bounded). Let x := {x₁, x₂, . . .} be a sequence in S.

A. Do you have enough information to tell if x is a convergent sequence? Explain or give an example illustrating your response.

B. Do you have enough information to tell if x has a convergent subsequence? Explain.  

C. Let x⁰ be a cluster point of x [x⁰ is a cluster point of the sequence x if there is a sub-sequence in x converging to x⁰ ]. Is x⁰ an element of S? Explain.

VII. Let A, B, C ⊂ R² where 经济学问题集代写

A := {(x, y) | y < x²};

B := {(x, y) | − 1 ≤ x ≤ 1, −1 ≤ y ≤ 2}; and

C := {(x, y) | − x + 2y = 1}.

A. Sketch each of these sets.

B. Of the three sets, A,B and C, which are closed?

C. Of the three sets, A,B and C, which are open?

D. Of the three sets, A,B and C, which are bounded?

E. Of the three sets, A,B and C, which are compact?

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