< b="" />Explanation: < />

In order to understand why the correct answer is option 'C', let's break down the given options and analyze each one.

< b="" />a) Bounded above: < />
A sequence is said to be bounded above if there exists a real number M such that every term in the sequence is less than or equal to M. However, being bounded above does not necessarily guarantee the existence of a convergent subsequence. For example, consider the sequence {1, 2, 3, 4, ...}. This sequence is bounded above by any number greater than or equal to 4, but it does not have a convergent subsequence.

< b="" />b) Bounded below: < />
Similar to being bounded above, a sequence is said to be bounded below if there exists a real number m such that every term in the sequence is greater than or equal to m. Again, being bounded below does not guarantee the existence of a convergent subsequence. For example, consider the sequence {-1, -2, -3, -4, ...}. This sequence is bounded below by any number less than or equal to -1, but it does not have a convergent subsequence.

< b="" />c) Bounded: < />
A sequence is said to be bounded if it is bounded above and bounded below. If a sequence is bounded, then it must have both an upper bound and a lower bound. This condition is necessary for the existence of a convergent subsequence. By the Bolzano-Weierstrass theorem, every bounded sequence in ℝ (the set of real numbers) has a convergent subsequence. Therefore, the correct answer is option 'C'.

< b="" />d) Monotone: < />
A sequence is said to be monotone if it is either increasing (each term is greater than or equal to the previous term) or decreasing (each term is less than or equal to the previous term). While monotone sequences do have convergent subsequences, not all convergent subsequences come from monotone sequences. Therefore, being monotone is not a sufficient condition for the existence of a convergent subsequence.

In conclusion, a sequence contains a convergent subsequence if and only if it is bounded (option 'C'). Being bounded above or bounded below (options 'A' and 'B') is not enough, and being monotone (option 'D') is not a necessary condition for the existence of a convergent subsequence.