A sequence cannot converge to more than one limit

A sequence is a list of numbers arranged in a specific order. The limit of a sequence represents the value that the terms of the sequence approach as the index approaches infinity. In other words, it is the value that the terms of the sequence get arbitrarily close to as the sequence progresses.

Definition of Convergence
A sequence is said to be convergent if it has a limit. In other words, a sequence converges if its terms get arbitrarily close to a specific value as the sequence progresses.

Definition of Boundedness
A sequence is said to be bounded if all its terms lie within a certain range. In other words, there exists a positive number M such that the absolute value of each term of the sequence is less than or equal to M.

Explanation of the Statements

a) A sequence cannot converge to more than one limit
- This statement is true. A sequence can only converge to a single limit. If a sequence has multiple limits, it means the terms of the sequence do not approach a specific value as the sequence progresses.

b) Every bounded sequence is convergent
- This statement is not true. A bounded sequence does not necessarily converge. A sequence can be bounded, meaning all its terms lie within a certain range, but it may not approach a specific value as the sequence progresses. For example, consider the sequence (-1, 1, -1, 1, -1, 1, ...). This sequence is bounded between -1 and 1, but it alternates between these two values and does not converge to a single limit.

c) Every bounded sequence is not convergent
- This statement is true. A bounded sequence may or may not converge. A sequence can be bounded but not converge, as explained in the previous statement (b).

d) Limit of a sequence is unique
- This statement is true. The limit of a sequence is unique. If a sequence converges, it approaches a specific value, and that value is the limit of the sequence. There cannot be multiple limits for a convergent sequence.

Conclusion
Out of the given statements, option 'C' is not true. Every bounded sequence may or may not converge. The other statements are true, with option 'A' stating that a sequence cannot converge to more than one limit, option 'B' stating that every bounded sequence is not necessarily convergent, and option 'D' stating that the limit of a sequence is unique.