What are the assumption(s) and conclusion(s) required to prove the statement directly? Prove, using the definition of a Cauchy sequence, that the following is a Cauchy sequence.

Solving practice problems of math 330

Calculus

Exam 1 – Math 330, Fall 2022

Each of the 5 problems will be graded out of 6 points.

Limits

(a) Complete the definition. We say that the limit of the sequence {an} is L ∈ R and

write limn→∞

an = L iff …

(b) Find L and prove, using the definition of a limit, that

limn→∞

2 + 3n

2n + 1

= L.

(a) Complete the definition. We say that the sequence {an} is a Cauchy sequence iff

(b) Prove, using the definition of a Cauchy sequence, that the following is a Cauchy sequence

4

3n − 2

n=1

Negations

(a) Complete the definition: we say {an} is a bounded sequence…

(b) Negate the definition above to say what we mean by {an} is not a bounded sequence (unbounded sequence)…

(c) The following statement is false. Negate the statement. Prove the negation is true (i.e. present a counterexample to the original statement).

∀n ∈ N, if 5n − 4 is even, then n is odd.

Consider the following: ∀x ∈ R, if −8 < x < 0, then −

16

8x + x

2

≥ 1

(a) What are the assumption(s) and conclusion(s) required to prove the statement directly?

(b) Prove the statement.

True or False. Circle T for true or F for false. In each of the following, suppose {an} represents an arbitrary sequence and L is an arbitrary real number.

T F If limn→∞

an = L, then the sequence {an} is bounded and it is a Cauchy sequence.

T F ∃M ∈ R, ∀N ∈ N, N < M.

T F Suppose x ∈ R. If x ≥ 0, then ∃ϵ ∈ R such that (x − ϵ, x + ϵ) ⊂ [0,∞).

T F Suppose a ∈ R and b > 0. We have |a| < b iff −b ≤ −a or a ≥ b

T F If {an} is unbounded, then limn→∞

an ̸= L.

T F Suppose limn→∞

bn = B and limn→∞

cn = C ̸= 0. Then

limn→∞

bncn = BC and limn→∞

bn

cn

=

B

C

 

What are the assumption(s) and conclusion(s) required to prove the statement directly? Prove, using the definition of a Cauchy sequence, that the following is a Cauchy sequence.
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