Final answer:
Any subsequence of a bounded sequence must also be bounded, as it consists of elements from the original sequence that are within the established bounds.
Step-by-step explanation:
If a sequence is bounded, it means that there is a real number that acts as an upper limit above which no elements of the sequence can go, and a real number that acts as a lower limit below which no elements of the sequence can fall. Given this definition, we can deduce that any subsequences extracted from the original sequence must also be bounded, as they are composed of elements from the original sequence which is itself bounded.
A subsequence is created by taking elements from the original sequence, without changing their order, but not necessarily including every element. Because the original sequence has a highest and a lowest value (upper bound and lower bound), every element selected to be part of a subsequence will also lie within these bounds. Therefore, every subsequence will inherit the property of being bounded from the original sequence.
For example, if you have a sequence that alternates between 1 and 2, like (1, 2, 1, 2, ...), it's bounded between 1 and 2. Any subsequence you choose, such as (1, 1, ...) or (2, 2, ...), is also bounded between 1 and 2.