University at Albany Introduction to Analysis Limit Definition Questions Real analysis questions Real analysis questions Real analysis questions Real analy

University at Albany Introduction to Analysis Limit Definition Questions Real analysis questions Real analysis questions Real analysis questions Real analysis questions Real analysis questions Real analysis questions en
t-3
Introduction to Analysis — IAUSS 2020
Assignment-3
Due: TBD
1. Let f : R → R and let c ∈ R. Show that lim f (x) = L if and only if lim f (x + c) = L
x→c
x→0
m
2. (a) Use either the − δ definition of limit or the Sequential Criterion for limits, to establish
the following limits

1
x+5
= −1
=4
iii. lim
i. lim
n→2
n→−1
1−x
2x + 3
2
2

x −x+1
x − 3x
1
ii. lim
iv. lim
=
=2
n→1
n→6
x+1
2
x+3
3. (a) Prove that lim cos(1/x) does not exist but that lim x cos(1/x) = 0
x→0
x→0
(b) Let f, g be defined on A to R and let c be a cluster point of A
i. Show that if both lim f and lim(f + g) exist, then lim g exists.
x→c
ig
n
x→c
x→c
ii. If lim f and lim f g exist, does it follow that lim g exists?
x→c
x→c
x→c
(c) Suppose that f and g have limits in R as x → ∞ and that f (x) ≤ g(x) for all x ∈ (a, ∞).
Prove that lim f ≤ lim g
n→∞
n→∞
4. (a) Let K > 0 and f : R → R satisfy the condition |f (x) − f (y)| ≤ K|x − y| for all x, y ∈ R.
Show that f is continuous at every point c ∈ R.
As
s
(b) Let f, g be defined on R and let c ∈ R. Suppose that lim f = b and that g is continuous
x→c
at b. Show that lim(g ◦ f ) = g(b).
x→c
5. (a) If f (x) = x and g(x) = sin(x), show that both f and g are uniformly continuous on R but
that their product f g is not uniformly continuous on R.
(b) Show that if f is continuous on [0, ∞) and uniformly continuous on [a, ∞) for some
positive constant a, then f is uniformly continuous on [0, ∞).
Introduction to Analysis IAUSS
2020(Summer)
Test
First Name
ST
Last Name
Question#
1
2
3
4
5
Total
Marks Available
18
24
16
24
18
100
Marks Obtained
INSTRUCTIONS
TE
1. Before the examination is scheduled to begin, TURN OFF YOUR CELL PHONE
2. The questions are to be answered in the spaces provided. Show details of all solutions.
3. This exam has SIX pages, including this one.
4. In no circumstances may the candidate take this exam from the examination room.
5. Candidates must not have in their possession books, notes or papers of any kind, nor any portable
electronic devices.
6. SHOW YOUR ALL WORK to get a full credit.
–2–
Marks
[18]
1. Find the supremum and infimum of each S. State whether they are in S
(a) S = {x|x = −(1/n) + [1 + (−1)n ]n2 , n ≥ 1}
(b) S = {x|x2 ≤ 7}
(c) S = {x||2x + 1| < 5} –3– [24] 2. (a) Prove that a set T1 is denumeratble if and only if there is a bijection from T1 onto a denumerable set T2 . (b) Prove in detail that if S and T are denumerable, then S ∪ T is denumerable. (c) Prove that the collection F (N) of all finite subsets of N is countable. –4– [16 ] 3. (a) Let x1 > 1 and xn+1 = 2 −
limit
1
for n ∈ N. Show that (xn ) is bounded and monotone. Find the
xn
(b) Let A be an infinite subset of R that is bounded above and let u = supA. Show there exists an
increasing sequence (xn ) with xn ∈ A for all n ∈ N such that u = lim (xn ).
n→∞
–5–
[24
4. Use either the − δ definition of limit or the Sequential Criterion for limits, to establish the following
limits

1
(a) lim
= −1
n→2 1 − x

2
1
x −x+1
=
(b) lim
n→1
x+1
2

x+5
(c) lim
=4
n→−1 2x + 3
2

x − 3x
=2
(d) lim
n→6
x+3
–6–
[18]
5. (a) If f (x) = x and g(x) = sin(x), show that both f and g are uniformly continuous on R but that
their product f g is not uniformly continuous on R.
(b) Show that if f is continuous on [0, ∞) and uniformly continuous on [a, ∞) for some positive
constant a, then f is uniformly continuous on [0, ∞).
[100]

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