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# 经济学问题集代写 Economics 426代写 经济学代写 经济学代做

159

## Economics 426: Problem Set 5

Points Possible: 100

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

### 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.

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 RN as follows: Let A ⊂ RN , B ⊂ RN . 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 xn 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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