Sequences

Question 1

A sequence is?

Answer 1

A function whose domain is the set of positive integers.

Question 2

A sequence (an) converges to a real number A if and only if?

Answer 2

Foe each €>0 there is a positive integer N such that for all n>=N we have |an - A| < €.

Question 3

A set Q of real numbers is a neighborhood of a real number x if and only if?

Answer 3

Q contains an interval of positive length centered at x. That is there is €>0 such that (x-€,x+€) subset Q.

Question 4

If a neighborhood of A contains all but a finite number of term of sequence (an) then?

Answer 4

The sequence (an) converges.

Question 5

If (an) converges to A and to B then?

Answer 5

A=B.

Question 6

A sequence is bounded from above if and only if?

Answer 6

There is a real number M such that an<=M for all n.

Question 7

A sequence is bounded from below if and only if?

Answer 7

There is a real number P such that P<=an for all n.

Question 8

A sequence is bounded if?

Answer 8

It is bounded from above and below.

Question 9

If (an) is bounded then?

Answer 9

(an) converges to A.

Question 10

A sequence (an) is said to be Cauchy if and only if?

Answer 10

For each €>0, there is a positive integer N such that n,m >=N, then |an - am| <€.

Question 11

Let S be a set of real numbers. A real number A is an accumulation point if and only if?

Answer 11

Every neighborhood of A contains infinitely many points of S.

Question 12

Every real number is an accumulation point of the?

Answer 12

Set of rational numbers. Also the set of irrational numbers.

Question 13

Every bounded infinite set of real numbers has?

Answer 13

At least one accumulation point.