S1, C7: Convergent sequences

Question 1

Let x be a real number. A sequence of real numbers

a0, a1, a2, . . . is said to converge to x if we have

Answer 1

for all ε>0, there exists a natural number N s.t. for all n > N
|an - x|

Question 2

Sequences converging to two different real numbers

Answer 2

A sequence a0, a1, . . . cannot converge to two different

real numbers x and y.

Question 3

What is the limit of an = n-1/n

Answer 3

1.

Question 4

Suppose that a0, a1, . . . and b0, b1, . . . are sequences, and suppose the sequence ai converges to a, and the sequence bi converges to b.
If ai

Answer 4

a

Question 5

Let a0, a1, . . . be a sequence that converges to x, and let b0, b1, . . . be a sequence that converges to y. Then the sequence a0 + b0, a1 + b1, . . . converges to

Answer 5

x+y

Question 6

(Sandwich Lemma) Suppose we have three sequences: a0, a1, a2, . . .,
and b0, b1, b2, . . . and c0, c1, c2, . . ., such that:
(i) the sequences ai and ci both converge to the same number x;
and
(ii) for all i we have
ai

Answer 6

x also.

Question 7

What is a Cauchy sequence?

Answer 7

A sequence a0, a1, a2, . . . is said to be Cauchy if
for all ε > 0, there exists a natural number s.t. for all m, n >N,
| am- an|

Question 8

Which convergent sequences are cauchy?

Answer 8

all.