Real Analysis Definition and Explanation

[Mr Davis 97]

Homework Statement

1) Suppose $t$ is a subsequential limit for $(s_n)$. Write the precise definition of the meaning of this statement.

2) Explain why there exists a strictly increasing sequence $(n_k)^\infty_{k=1}$ of natural numbers such that $\lim s_{n_k}=t$.

Homework Equations

The Attempt at a Solution

I am not exactly sure what to do for 1). Doesn't this just mean that there exist a subsequence that converges to t? I am not exactly sure how precisely I need to state it, or how I would if a lot of precision is necessary.

For 2), doesn't this just follow from the definition of a subsequential limit?

 

[Eclair_de_XII]

For 2), doesn't this just follow from the definition of a subsequential limit?

Is it mentioned that the sequence $\{s_n\}_{n=1}^\infty$ is bounded?

 

[Mr Davis 97]

Is it mentioned that the sequence $\{s_n\}_{n=1}^\infty$ is bounded?

Is is not mentioned that the sequence is bounded or unbounded.

 

[Mark44]

If the sequence is $\{s_n\}$, then a subsequence is written as $\{s_{n_k}\}$, where $n_1, n_2, n_3, \dots,$ is an increasing sequence of indices. Can you write a definition of a subsequence that converges to t using this notation?

 

[member 587159]

Is it mentioned that the sequence $\{s_n\}_{n=1}^\infty$ is bounded?

This isn't relevant for the problem. The statement is true with or without boundedness.

The Attempt at a Solution

I am not exactly sure what to do for 1). Doesn't this just mean that there exist a subsequence that converges to t? I am not exactly sure how precisely I need to state it, or how I would if a lot of precision is necessary.

For 2), doesn't this just follow from the definition of a subsequential limit?

(1) Write it as formally as you can!
(2) We can't help you with (1) if you don't tell us what definition of subsequential limit you use.

 

[Stephen Tashi]

Homework Statement

1) Suppose $t$ is a subsequential limit for $(s_n)$. Write the precise definition of the meaning of this statement.

Since you've recently asked questions about the logic of quantifiers, my guess is that this problem is testing your skill in writing definitions involving the use of "there exists".

2) Explain why there exists a strictly increasing sequence $(n_k)^\infty_{k=1}$ of natural numbers such that $\lim s_{n_k}=t$.

As @Mark44 indicated, if you write a definition that says "there exists a subsequence", that statement is not identical to the claim that "there exists a strictly increasing sequence $(n_k)$ of natural numbers such that...". , although the latter statement might be taken for granted in a typical mathematical article. So imagine someone insists that you "split hairs" and explain why a definition that says "there exists a subsequence..." implies a statement about the existence of a certain strictly increasing sequence of natural numbers. (To do that, you need to use the formal definition of a subsequence.)