Question

Let (an) be a sequence and t be a subsequential limit of (an). What is the relationship between lim inf an, t, and lim sup an?

A. Lim inf an < t < lim sup an.
B. Lim inf an > t > lim sup an.
C. Lim inf an = t = lim sup an.
D. The relationship cannot be determined.

Answer 1

The relationship between lim inf an, t, and lim sup an is that lim inf an < t < lim sup an.

Step-by-step explanation:

The relationship between lim inf an, t, and lim sup an can be described in terms of their magnitude.

First, let's define the terms:

• lim inf an: The limit inferior of the sequence (an) represents the smallest sub sequential limit of the sequence.
• lim sup an: The limit superior of the sequence (an) represents the largest subs sequential limit of the sequence.
• t: A sub sequential limit of the sequence (an) is a value that the sequence approaches as n tends to infinity.

The relationship between these values can be summarized as follows:

• If t lies between lim inf an and lim sup an, then we have lim inf an < t < lim sup an.

Therefore, the correct answer is A. Lim inf an < t < lim sup an.