Math 4101/5101

Practice Problems
 
1. Use the triangle inequality to show that a, b R,  |a| |b|  |a b|. hint: a = a b + b.
2. Use induction to show that for every positive integer n and x1 , x2 , . . . , xn R, |x1 + x2 + + xn | |x1 | +
|x2 | + + |xn |.
3. Fix x 0 , y0 R with y0 6= 0. n
> 0, find o 1 , 2 > 0 n
such that if 0 o< |x x0 | < 1 and 0 < |y y0 | < 2 , then
|y0 | |y0 |2
, 2 = min |y20 | , 4(1+|x
 x x0 
 y y0  < . Solution: 1 = min 1, 2 0 |)
will work.
Recall that lim f (x) = l means > 0, = > 0 such that if 0 < |x a| < , then |f (x) l| < , while
xa
lim f (x) = l means > 0, N = N > 0 such that if x > N , then |f (x) l| < .
x+
p 
|x| + 1 sin x1 = 0


4. Prove lim ln
x0
2
5. Prove lim e2(x +1)
= 0.
x+
6. Squeeze Theorem. Say that f, g, h are functions such that h(x) f (x) g(x) for all x in some interval containing
a (except possibly at a itself) and that lim h(x) = l and lim g(x) = l. Show that lim f (x) = l. Formulate and prove
xa xa xa
the lim version of this statement.
x+
1 1
7. Fix a 6= 0. Prove lim 2 = a2 .
xa x
8. Prove that if lim f (x) = l, then f is locally bounded near a. That is, , M > 0, such that |f (x)| < M for all x
xa
with 0 < |x a| < .
9. A function f is continuous at a point a if lim f (x) = f (a). Say that f satisfies f (x + y) = f (x) + f (y) for all x, y,
xa
and that f is continuous at 0. Prove that f is continuous at a, a R.
10. Prove that f is continuous at a, so is |f |.
11. Sequences. A sequence {an } of real numbers converges to a number l, i.e. lim an = l if > 0, N = N > 0
n
such that if n > N , then |an l| < . Prove that convergent sequences are bounded. That is, if lim an = l, then
n
M > 0, such that |an | M, n 1.
12. Prove that the product of convergent sequences is itself convergent. That is, if {an }, {bn } are convergent sequences
with lim an = l and lim bn = m, then lim an bn = lm.
n n n
13. Say {an } is a sequence of positive numbers with lim an = 0. Is it true that {an } is nonincreasing for n
n
sufficiently large, i.e. an+1 an for n large enough? Prove or provide a counterexample.
14. A function f is convex on an interval I if x, y I and 0 t 1, f ((1 t)x + ty) (1 t)f (x) + tf (y).
(a) Verify that the functions f (x) = |x|, g(x) = x2 are convex on R.
* (b) Prove that if f > 0 on I, then f is convex on I. Hint: Use F.T.O.C. and the fact that f is increasing.
(c) Verify that h(x) = ln x is convex on (0, ).
(d) Prove that if f is concave function on Rn , then the set C = {x Rn : f (x) > } is convex.
15. Prove that if C Rn is convex, then for any x1 , . . . , xm C and and nonnegative numbers t1 , t2 , . . . , tm with
t1 + t2 + + tm = 1, the convex combination t1 x1 + t2 x2 + + tm xm also lies in C. (Use induction on m.)
|lm|
16. Limits are unique. If lim f (x) = l, and lim f (x) = m, prove that l = m. Hint: if l 6= m, take = 2 in the
xa xa
definition of limit, and get a contradiction.
17. Prove that if {an } is monotone decreasing and bounded below, then {an } converges.
n o
18. Prove (using the definition) that the sequence 5(1)
n

n1
+ 3 is a Cauchy sequence.
19. Using only the definition of Cauchy sequence, prove that if {an } is a Cauchy sequence, then theres a constant
C such that |an | C, n 1.
20. Let A Rn . Prove that if x is a limit point of A, then theres a infinite sequence of points of A converging to x.
21. Fix r > 0 and x0 Rn . The (open) ball of radius r, centered at x0 is the set Br (x0 ) = {x Rn : |x x0 | < r}.
(a) Show that Br (x0 ) is an open subset of Rn .
(b) Show that Br (x0 ) is a convex subset of Rn . Recall a set A Rn is convex if for any x1 , x2 A, the straight
pP n 2
line segment (1 t)x1 + tx2 , 0 < t < 1 also lies in A. Notation: for x = (x1 , x2 , . . . , xn ) Rn , |x| = i=1 xi
(which we used to denote by kxk).
22. Prove that the interval (a, b) is an open subset of R.
23. Show that b is a limit point of (a, b) by explicitly finding a point y 6= b (b , b + ) (a, b), > 0.
24. Show that any point x satisfying |x x0 | = r is a limit point of Br (x0 ) by explicitly finding a point y 6= x
B (x) Br (x0 ), > 0.
25. Show that the set Z of integers has no limit points (hence is trivially closed).
26. For a subset A Rn , let A denote the set of all limit points of A, and let A denote the set of all interior points
of A. Recall A = A A
(a) Prove that A is closed.
(b) Prove that A and A have the same limit points.
(c) Prove that A is open.
(d) Prove that A is open if and only if A = A .
(e) Prove that A is the largest open set contained in A.
(f) Prove that (A )c = Ac .
27. Let A = [1, 1) {2}.
(a) Find A (the closure of A), and A (the interior of A).
(b) Is A closed?, convex?, bounded?, compact? (Recall a set A is bounded if C such that |x| C, x A.)
28. Let {Ak } n
k=1 be subsets of R .
(a) Prove that for any n, n Ak = n Ak
k=1 k=1
(b) Prove that
k=1 Ak k=1 Ak .
Show by example that this inclusion may be proper.

(c) Prove that
k=1 Ak Give an example of sets A, B,Twhere A B
k=1 Ak . 6= A B.

29. Give an example of an infinite number of open sets Gn , where Sn=1 Gn is not open.

30. Give an example of an infinite number of closed sets Fn , where n=1 Fn is not closed.
31. Fix x, y Rn . Prove that if z is on the straightline segment connecting x and y, then |x y| = |x z| + |z y|.
Recall, the straight line segment between a, b Rn is the set of all vectors of the form (1 t)a + tb, where t [0, 1].
32. (a) For bounded sequences {an }, {bn }, prove that

inf an + inf bn inf (an + bn ) inf an + sup bn sup(an + bn ) sup an + sup bn .

n n n n n n n n

(b) sup(an ) = inf (an )

n n

(c) For nonnegative sequences {an }, {bn }, sup(an bn ) sup an sup bn .

n n n
(d) | sup an sup bn | sup |an bn |. Hint: use the last inequality in (a)
n n n
3(1)n +sin( n+1)+5 (1)n n
33. Find lim sup and lim inf , for the sequences an = (1)n , bn = tan1 n, cn =
n
, dn = n+1 .
n n
34. Prove that (0, 1) is not a compact subset of R, by exhibiting an open cover which has no finite subcover.
35. Fix r > 0 and x0 Rn . The closure of the open ball of radius r, centered at x0 , is the set B r (x0 ) = {x Rn :
|x x0 | r}. Show that this set is closed in two ways:
(a) By proving that its complement, B r (x0 )c = {x Rn : |x x0 | > r} is an open subset of Rn .
(b) By proving that it contains all its limit points.
36. Fix 0 r1 < r2 and x0 Rn . The (open) annular region with center x0 between the discs Br1 (x0 ) and Br2 (x0 ),
is the set Ar1 ,r2 (x0 ) = {x Rn : r1 < |x x0 | < r2 }.
(a) Prove that Ar1 ,r2 (x0 ) is an open subset of Rn .
(b) Prove that Ar1 ,r2 (x0 ) = {x Rn : r1 |x x0 | r2 } is closed by showing that it contains all its limit points.