solving practice problems of math 330

Exam 1 – Math 330, Fall 2022
Each of the 5 problems will be graded out of 6 points.
1. 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.
2. (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
3. 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.
4. 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.
5. 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