Upload
eddiekimbowa
View
222
Download
0
Embed Size (px)
Citation preview
8/13/2019 Practice Problems Set1
1/3
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?
8/13/2019 Practice Problems Set1
2/3
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.
8/13/2019 Practice Problems Set1
3/3
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.