MATH 18.100A/18.

1001 Midterm MIT

Real Analysis
Midterm
October 16, 2020

The following exam consists of 5 problems worth 15 points each. Solutions should be
written in complete sentences where appropriate.
The midterm is open book, open notes, but collaborating with other students or the
internet is strictly prohibited. By signing your name below, you attest to following these rules
for the exam. Evidence to the contrary will be treated as academic misconduct and will responded
to according to MIT Institute Policy 10.2.

Name:

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MATH 18.100A/18.1001 Midterm MIT

1. (a) (5 points) Let f : A → B, and let C, D be subsets of B. Prove

f −1 (C ∩ D) = f −1 (C) ∩ f −1 (D).

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MATH 18.100A/18.1001 Midterm MIT

(b) (5 points) When E is a countable subset of R, is the complement R \ E always uncount-

able? Explain why or why not.

(c) (5 points) When E is an uncountable subset of R, is the complement R \ E always

countable? Explain why or why not.

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MATH 18.100A/18.1001 Midterm MIT

2. A subset of real numbers U ⊂ R is open if for all x ∈ U , there exists  > 0 such that
(x − , x + ) ⊂ U . A subset of real numbers F ⊂ R is closed if F c is open.

(a) (5 points) State what it means to say U is not open.

(b) (5 points) Prove that if U is not open, then there exists x ∈ U and a sequence {xn }n of
elements of U c such that

lim xn = x.
n→∞

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MATH 18.100A/18.1001 Midterm MIT

(c) (5 points) Suppose F ⊂ R has the following property: for every convergent sequence
{xn }n of elements of F we have limn→∞ xn ∈ F . Prove that F is closed.
Hint: Argue by contradiction using (b).

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MATH 18.100A/18.1001 Midterm MIT

3. (a) (5 points) Use the definition of convergence to prove that

10n2
lim = 10.
n→∞ n2 + 16n + 1

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MATH 18.100A/18.1001 Midterm MIT

(b) For each of the following scenarios, give an example satisfying the stated property. For-
mal proofs are not required, but some explanation may be useful.
(i) (5 points) A sequence {xn } converging to 0 which is not monotonic.

(ii) (5 points) An unbounded sequence that has a convergent subsequence.

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MATH 18.100A/18.1001 Midterm MIT

4. (a) Let {xn } and {yn } be bounded sequences of real numbers.

(i) (5 points) Prove that the sequence {xn + yn } is bounded.

(ii) (5 points) Prove that

lim sup (xn + yn ) ≤ lim sup xn + lim sup yn .

n→∞ n→∞ n→∞

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MATH 18.100A/18.1001 Midterm MIT

(b) (5 points) Let E denote the set of all real numbers in (0, 1) with decimal expansion
involving only 1’s and 2’s:

E = {x ∈ (0, 1) : ∀j ∈ N, ∃d−j ∈ {1, 2}, such that x = 0.d−1 d−2 . . .}.

Note that 0.2 ∈

/ E but 0.222222 . . . ∈ E. Prove that 0.1111111 . . . is a cluster point of
E.

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MATH 18.100A/18.1001 Midterm MIT

5. (a) (5 points) Suppose that for all n ∈ N, an > 0, bn > 0 and

an
lim = L > 0.
n→∞ bn
P P
Prove that an converges if and only if bn converges.

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MATH 18.100A/18.1001 Midterm MIT

(b) Find all real numbers x such that the series converges. Find all real numbers x such
that the series converges absolutely.
(i)
∞
X (−1)n
(x − 10)n
2020n
n=0

(ii)
∞
X
n!xn!
n=0

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18.100A / 18.1001 Real Analysis

Fall 2020

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