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Real Analysis，please check the file that I uploaded，My grade is 87.5% now，so I need 90%+ support tomorrow to get an A in this class，please check the file that I u’puploaded，It was our midterm，It should be very similar. Please make sure you are familiar with Real Analysis, It should be easy and finished around 1-1.5hour ，thanks.

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Math 370 Midterm Exam

March 18, 2021

There are four questions on the exam. Each question is worth ten points.

1. Do TWO of the following:

a) What does it mean for a sequence to be Cauchy?

b) State the Bolzano-Weierstrass Theorem.

c) What does it mean for a sequence {xn } to converge to a real number a?

d) State the Squeeze Theorem for sequences.

2. Are the following true or false? Justify your answers briefly.

a) If xn > 0 for all n ∈ N , and xn → x, then x > 0.

b) If {xn } is a sequence such that {x2n } converges, then {xn } converges.

c) If {xn } is Cauchy, then it is increasing.

d) For any positive number r, {rn } converges.

3. Suppose xn → a, and {yn } is a sequence with the property that |xn −yn | ≤

for all n ∈ N . Show that yn → a.

1

1

n

4. Let

f (x) =

3 sin x

.

x2 + 1

Show, using the definition, that limx→0 f (x) = 0. (HINT: x2 + 1 ≥ 1 and

| sin x| ≤ |x| for all real numbers x.)

2

Final Exam, Math 370

There are five questions on the exam. Each question is worth ten points.

1. Do TWO of the following:

a) State the Mean Value Theorem.

b) What does it mean for a function f to be integrable on [a, b]?

c) What does it mean for a function f : R → R to be uniformly continuous?

d) State the Intermediate Value Theorem.

2. Are the following true or false? Justify your answers briefly.

a) Let f be continuous on [0, 1]. Then

R1

0

f (x)dx ≥ 0.

b) Let f, g : (0, 1) → R be such that f (x) < g(x) for all x ∈ (0, 1). If limx→0+ f (x) = L,
and limx→0+ g(x) = M , then L < M .
c) Let f be continuous on (0, 1). If {xn } in (0, 1) is a Cauchy sequence, then {f (xn )} is
a Cauchy sequence.
d) If f is bounded on [0, 1], then f is integrable on [0, 1].
3. a) Define what it means for a sequence {an } to converge to 0.
b) Show carefully that if an → 0, then a2n → 0. (If you are using theorems from the book
here, please state them.)
4. a) Define what it means for limx→0+ f (x) = 0.
Now let
1
.
f (x) = x cos
x
b) Show, using an epsilon-delta proof, that limx→0+ f (x) = 0.
c) Conclude that f is uniformly continuous on (0, 1).
5. Let f : [0, 2] → R be a differentiable function such that f (0) = f (2) = 0, and |f 0 (x)| ≤ 1
for all x ∈ [0, 2].
a) Use the Mean Value Theorem to prove that |f (t)| ≤ t for all t ∈ [0, 1].
b) Use a similar argument to prove that |f (t)| ≤ 2 − t for all t ∈ [1, 2].
c) Conclude that
Z 2
0
f (t)dt ≤ 1.
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Tags:
value theorem
Continuous function
Squeeze Theorem
differentiable function
uniformly continuous
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