Practice Problems Set1

8/13/2019 Practice Problems Set1

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MAA 5228 and MAA 4226 Introductory Analysis I (Fall 2013)

Practice Problems Set 1

1. Let Abe subset ofR which is bounded from above. Prove that sup(A) is either an element ofA ora limit point ofA. State and prove a similar result for inf(A).

2. Let dbe a metric on the set X. Define

d(x, y) = min{d(x, y), 1} for allx, y X .

Prove that d is also a metric on X. Is d a metric if min is replaced by max? Prove your conclusion.

3. Let{xn}be a convergent sequence in a metric space. Suppose thatx is the limit of{xn}. Prove thatthe set A= {xn} {x} is a closed set.

4. Let X be a metric space. Suppose that every bounded infinite set has a limit point. Prove thatXis complete.

5. Given four metric spaces [0, 1], R, Q, and X, where Q is the set of rational numbers equipped with the

usual metric, and X is an infinite set equipped with the discrete metric. Which ones are complete?Which ones are compact? Which ones are connected?

6. Let Aand B be subsets of a metric space X. Prove that

(A B) =A B,

whereA denotes the set of all limit points of the set A.

7. Let Abe a subset of a metric space X. Prove that

bd(A) \ A= A \ A,

where bd(A) denotes the boundary ofA, which is the set of all boundary points ofA.

8. Let Aand B be subsets of a metric space X. Prove that

A B=A B.

9. Consider the n-dimensional Euclidean space Rn. Let x Rn and > 0 and A = B(x, ). Findint(A), A, A, and bd(A). Justify your answers.

10. Consider the discrete metric space (X, d). Let x X and >0 and A= B(x, ). Find int(A), A,

A, and bd(A). Justify your answers.11. LetXbe a nonempty set. Let d1and d2be metrics onX. Suppose that there exist positive constants

and such that d1(x, y) d2(x, y) d1(x, y) for all x, y X. Prove that a subsetU ofX isopen in the metric space (X, d1) if and only if it is open in the metric space (X, d2). Is this true forclosed sets? And Why?


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12. Letm be a positive integer. Equip Rm with the standard metric. Suppose thatA and B are compactsubsets ofRm.(a) Prove that their sum A + B is a compact subset ofRm, where

A + B= {x + y: x A, y B}.

(b) Prove that their product A B is compact in R2m, where

A B = {(x, y) :x A, y B},

and (x, y) denotes the point (x1, x2, , xm, y1, y2, , ym) which is a points in R2m.

13. Let A be a set in a metric space. If A is connected and contains more than one point, show thatevery point ofA is a limit point ofA.

14. Let A= {(x, y) R2 : x4 + y4 = 1}. Show that A is compact. Is A connected? Why?

15. Prove that there exists a sequence n1 < n2 < n3 < of positive integers such that the limitlimk

sin(nk) exists.

16. Let f : R R be continuous. Prove thatf maps bounded sets into bounded sets, that is, if A is

a bounded subset ofR, then f(A) is a bounded subset ofR. Is this true for a continuous functionthat maps Rn into Rm?

17. Let Aand B be open subsets of a metric space X such that A B =. Prove that A B = andAB= . Find an example of two open subsets CandDofR such that CD= but CD=.

18. Let (X, d) be a metric space and let x Xand Aa nonempty subset ofX. The distance betweenxand Ais defined by

d(x, A) = inf{d(x, y) : y A}.

(a) Prove that fis a continuous function from X into R.(b) Prove that d(x, A) = 0 if and only ifx A.

19. Let A and B be two nonempty sets in a metric space such that A is compact and B is closed, andthat A B = . Prove that d(A, B) > 0, where d(A, B) denotes the distance between A and Bdefined by

d(A, B) = inf{d(x, y) : x A, y B}.

20. Let Abe a connected subset ofR2. Define

B= {x R : (x, y) A for some y R}.

Prove that B is a connected subset ofR.

21. Let {xn} be a convergent sequence in a metric spaceX. Suppose thatx = limnxn. For every positiveinteger n, let Fn={xk : k n} {x}. (a) Show that all Fn are compact and Fn+1 Fn. (b) Find

n=1

Fn.

22. Let (X, d) be a metric space. Suppose that F is nonempty and compact and G is open in X suchthat F G. Prove that there exists a > 0 such that d(x, F) for all x Gc, where d(x, F)denotes the distance from xto F and Gc =X\ G.


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23. Let A be a connected set in a metric space X. Let B be any subset of X such that A B A.Prove that B is also connected. In particular, A is connected.

24. Let Aand B be two subsets ofRsuch that A B is a compact subset ofR2, where

A B= {(x, y) : x A, y B}.

Prove that both Aand B are compact subsets ofR.

25. Let A and B be two subsets ofR such that A B is a connected subset ofR2. Prove that both Aand B are connected subsets ofR.

26. Let Aand B be two subsets ofR such that A B is an open subset ofR2. Prove that both AandB are open subsets ofR.

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Practice Problems Set1