Question

Let (an) be a sequence of real numbers. Then there exists a subsequence of (an) (ank) such that lim ank as n goes to infinity = ______ := lim an as n goes to infinity. Similarly, ______ is also a subsequential limit of (an).

A. Lim inf an; lim sup an.
B. Lim sup an; lim inf an.
C. Lim an; lim inf an.
D. Lim an; lim sup an.

Answer 1

In a sequence, a subsequence is obtained by selecting some terms from the original sequence in a specific order. The limit inferior and limit superior of a sequence are the values that the sequence approaches as n goes to infinity. The correct answers are: A. Lim inf an; lim sup an.

Step-by-step explanation:

Let (an) be a sequence of real numbers. In a sequence, a subsequence is obtained by selecting some terms from the original sequence in a specific order. The given question asks about the existence of a subsequence (ank) such that the limit of (ank) as n goes to infinity is equal to the limit of the original sequence (an) as n goes to infinity.

The correct answers are: A. Lim inf an; lim sup an. The limit inferior (lim inf) and limit superior (lim sup) of a sequence are the values that the sequence approaches as n goes to infinity. The lim inf an and lim sup an are subsequential limits of the sequence.