The question seems to be incomplete. However, I can provide some general information about bounded sequences in R (the set of real numbers) and their properties.

A sequence (xn) in R is said to be bounded if there exists a real number M such that |xn| ≤ M for all n in natural numbers. This means that all the terms of the sequence are within the interval [-M, M].

Now, if x* is a number in R, there could be several statements about x* that could be equivalent in the context of the bounded sequence (xn). For example:

1. x* could be a limit point (or accumulation point) of the sequence. This means that for any positive number ε, there are infinitely many terms xn within the distance ε from x*.

2. x* could be the limit of a subsequence of (xn). This means that there exists a subsequence (xnk) of (xn) such that the limit of (xnk) as k approaches infinity is x*.

3. x* could be the supremum (least upper bound) or infimum (greatest lower bound) of the set of all limit points of the sequence (xn).

These are just examples and the actual statements would depend on the specific properties of the sequence (xn) and the number x* that you are considering.

Question

The question seems to be incomplete. However, I can provide some general information about bounded sequences in R (the set of real numbers) and their properties.

A sequence (xn) in R is said to be bounded if there exists a real number M such that |xn| ≤ M for all n in natural numbers. This means that all the terms of the sequence are within the interval [-M, M].

Now, if x* is a number in R, there could be several statements about x* that could be equivalent in the context of the bounded sequence (xn). For example:

1. x* could be a limit point (or accumulation point) of the sequence. This means that for any positive number ε, there are infinitely many terms xn within the distance ε from x*.

2. x* could be the limit of a subsequence of (xn). This means that there exists a subsequence (xnk) of (xn) such that the limit of (xnk) as k approaches infinity is x*.

3. x* could be the supremum (least upper bound) or infimum (greatest lower bound) of the set of all limit points of the sequence (xn).

These are just examples and the actual statements would depend on the specific properties of the sequence (xn) and the number x* that you are considering.

Knowee AI · Accepted Answer

The question seems to be incomplete. However, I can provide some general information about bounded sequences in R (the set of real numbers) and their properties.

A sequence (xn) in R is said to be bounded if there exists a real number M such that |xn| ≤ M for all n in natural numbers. This means that all the terms of the sequence are within the interval [-M, M].

Now, if x* is a number in R, there could be several statements about x* that could be equivalent in the context of the bounded sequence (xn). For example:

1. x* could be a limit point (or accumulation point) of the sequence. This means that for any positive number ε, there are infinitely many terms xn within the distance ε from x*.

2. x* could be the limit of a subsequence of (xn). This means that there exists a subsequence (xnk) of (xn) such that the limit of (xnk) as k approaches infinity is x*.

3. x* could be the supremum (least upper bound) or infimum (greatest lower bound) of the set of all limit points of the sequence (xn).

These are just examples and the actual statements would depend on the specific properties of the sequence (xn) and the number x* that you are considering.

If (xn) is a bounded sequence in R then the following statements about a numberx∗ ∈ R are equivalent

Question

If $(x_n)$ is a bounded sequence in $\mathbb{R}$ then the following statements about a number $x^* \in \mathbb{R}$ are equivalent:

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