The set of limit points of a bounded sequence isa)unboundedb)boundedc)...
Answer:
To understand why the correct answer is option 'B' (bounded), let's first define what a bounded sequence is and what a limit point is.
A bounded sequence is a sequence in which all the terms are within a certain range, or in other words, there exists a number M such that all the terms of the sequence are less than or equal to M in absolute value.
A limit point of a sequence is a point that can be approached infinitely close by the terms of the sequence. In other words, for any positive epsilon, there exists a term of the sequence that is within epsilon distance of the limit point.
Now, let's consider a bounded sequence. Since the sequence is bounded, there exists a number M such that all the terms of the sequence are less than or equal to M in absolute value. This means that the terms of the sequence are contained within a certain range.
The set of limit points of a bounded sequence is bounded.
When we consider the set of limit points of a bounded sequence, we are looking at all the points that can be approached infinitely close by the terms of the sequence. Since the terms of the sequence are bounded, they cannot escape beyond the range defined by the bound. Therefore, the set of limit points, which includes all the points that can be approached infinitely close by the terms of the sequence, is also bounded.
To illustrate this, let's consider a simple example. Suppose we have a bounded sequence {a_n} where -1 ≤ a_n ≤ 1 for all n. The terms of this sequence are contained within the range [-1, 1]. Now, let's say there is a limit point L. Since L is a limit point, for any positive epsilon, there exists a term of the sequence that is within epsilon distance of L. However, since the terms of the sequence are bounded, they cannot escape beyond the range of [-1, 1]. Therefore, the set of limit points of this bounded sequence is also bounded.
In conclusion, the set of limit points of a bounded sequence is bounded. Therefore, the correct answer is option 'B'.