MATH 142A MIDTERM 1

Monday, January 27, 2020 (50 minutes).

Please turn all cell phones o↵ completely and put them away.No books, notes, or electronic devices are permitted during this exam.Generally, you must show your work to receive credit.

• The back side of test pages will not be evaluated. You can use it for your owncomputations, but whatever you write there will not be evaluated.

Name (print):

Student ID number:

Question 1 three parts 25 points

Question 2 two parts 25 points

Question 3 two parts 25 points

Question 4 one part 25 points

Bonus Question one part 25 points

Total 4 questions 100 points

SOLUTIONS

1. [25 pts] Answer the following questions.1.1[10 pts] Given a set S ⇢ R, define inf S.

1.2[10 pts] Let S := {x 2 Q : x �p2}. Prove that there exists inf S 2 R and that it is

equalp2.

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bexVxES a b

X's R V x e S re is a lover lol fab Shldblow

by cordlay to completeness oeiom Fmfs e IR

N x ra Vxes V2 is lower lol

4 Suppose ly conkaldion Fb K att xs.ba

Bydenseness of ni R F re Q att b r R2

resondrcts E bekOr

1.3[5 pts] Prove that S has no minimum.

Suppose ly contoldion F a mmSen then

mins mfs ra A minsESEels

Enrico rufa

2. [25 pts] Answer the following questions.2.1[10 pts] Write what it means that a sequence (an)n converges.

2.2[15 pts] Prove that the sequence an = (�1)n

2 doesn’t converge.

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Supper ly contradiction 3 linin an a e IR then letE I so and let No it n N fan IIII itIt n nu N then I HI HII I On duty

E Iam al t la anI L th t ta z E

3. [25 pts] Solve (i) and (ii).3.1[15 pts] Prove that if (an)n, (bn)n and (cn)n are sequences such that limn an = x = limn cnand an bn cn for every n 2 N, then limn bn = x.

3.2[10 pts] Prove that limn(�1)n

n2 = 0.

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n N Ibn XI L meeYanH 1cm xl LE

It an In and cristathen

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4. [25 pts] Prove that the limit of a convergent sequence is unique.

Suppose ly contodelion Fkn bite s.tt

bin an b and lenin Knac

Let e 1k then FN Mx att

n N I am b c lbftn Nz I en de lbzI

Let Namath Nel ther n N

lb ol lb ansan cle lb antt Ian ol

theft lbft lb L l gC u

5. [25 pts] (Bonus Problem) Let (an)n be a sequence of integers, in other words an 2 Zfor all n � 1. Suppose that (an)n converges. Prove that (an)n is eventually constant: thatis, there is z 2 Z and N 2 N such that an = z for all n � N .

Prooflycentilton VNHN Fm n NattAm 4 Dmz

then nine an ou ly def att e E FNeIN sa

n N Ian all kI Ee Iden and Ian at a an I

Shiu an on E Ian at the andIananEZ

Liza