91Ó°ÊÓ

When studying convergent sequences, one important property is the uniqueness of the limit. This concept states that a sequence can have at most one limit. To explore this further, consider a sequence such as \( \{a_n\} \). If it converges, it means that as \( n \) becomes very large, the terms \( a_n \) get arbitrarily close to a certain value, which we refer to as the limit. If, theoretically, a sequence had two different limits, \( L \) and \( M \), we would have a contradiction. Why? Because according to the definition of convergence, for any tiny difference \( \epsilon \), the terms of the sequence would be within \( \epsilon \) distance from both \( L \) and \( M \) at the same time for large enough \( n \). But this would ultimately imply that \( L \) and \( M \) themselves are very close, and eventually, it turns out they must be the same. Thus, a convergent sequence cannot have two different limits.

The triangle inequality is a crucial mathematical tool that is often used in proofs involving limits and convergent sequences. In simple terms, the triangle inequality states that for any real numbers \( a \), \( b \), and \( c \), the direct path from \( a \) to \( c \) is shorter or equal than taking a detour through \( b \). Mathematically, this is expressed as \( |a - c| \leq |a - b| + |b - c| \).
This inequality helps when dealing with the uniqueness of limits. Imagine two possible limits \( L \) and \( M \) for a sequence \( \{a_n\} \). The triangle inequality allows us to write:

• \( |L - M| \leq |L - a_n| + |a_n - M| \)

Using this relationship, if both \( |L - a_n| \) and \( |a_n - M| \) can be made arbitrarily small (as determined by convergence), then \( |L - M| \) can too. Therefore, if \( L \) and \( M \) were two different limits, \( L - M \) would still need to be a very small number, effectively zero, forcing \( L \) to equal \( M \). This supports the conclusion of the uniqueness of limits.

Understanding convergence is key when working with limits of sequences. A sequence \( \{a_n\} \) is said to converge to a limit \( L \) if, no matter how small a number \( \epsilon \) you choose, there is some point in the sequence beyond which every term is \( \epsilon \)-close to \( L \).
More formally, for every positive real number \( \epsilon \), there exists a positive integer \( N \) such that for all \( n > N \), the inequality \( |a_n - L| < \epsilon \) holds.

• This means that after a certain point, all terms of the sequence get as close as we want to the limit \( L \).
• The larger \( n \) becomes, the closer \( a_n \) is to \( L \).

This definition is fundamental in demonstrating why sequences have unique limits. If a sequence were to have two limits, \( L \) and \( M \), it would imply that the terms approach both \( L \) and \( M \) as \( n \) grows. However, this is impossible under the definition, which ensures that a sequence converges to one specific point. Thus, a sequence can't converge to two different values, protecting the uniqueness of the limit.