91Ó°ÊÓ

In mathematics, when we say that a sequence has a limit, we mean that the terms of the sequence approach a specific value as the sequence progresses indefinitely. For a sequence \(a_n\), if the terms get closer and closer to a number \(a\) as \(n\) (which represents the position of the term in the sequence) increases, we write this as \(a_n \rightarrow a\).
This notation \(\rightarrow\) implies that the difference between each term \(a_n\) and \(a\), the limit, becomes arbitrarily small as \(n\) becomes very large. To grasp this better, think of being on a path that leads directly to a target. The further along the path you travel, the closer you get to your destination, until practically you are there.

• Approaching a number: The sequence terms steadily move closer to the limit.
• Consistent reduction in distance: The difference between \(a_n\) and \(a\) decreases continually.

A sequence is called convergent if it has a limit; that is, it approaches a specific number as it progresses through its terms. In our exercise, two sequences are involved, \(a_n\) converging to \(a\) and the sequence of differences \(b_n - a_n\) converging to 0.
Convergence is crucial because it confirms the behavior of the entire sequence over time, ensuring consistency in reaching the limit. It is a property that helps us predict where the sequence will ultimately end up.
For a sequence to be convergent:

• It must have a definable limit.
• The difference between terms and the limit becomes negligible over time.
• There should be a point beyond which all terms remain arbitrarily close to the limit.

The convergent nature of \(b_n\) in the exercise shows its progression towards \(a\), confirming its stable behavior as \(b_n = a_n + (b_n - a_n)\) converges.

The concept of the difference of sequences comes into play when examining the relationships between two sequences. In the exercise, the sequence difference \(b_n - a_n\) is critical to understanding the convergence of \(b_n\).
In simple terms, the sequence \(b_n - a_n\) measures how far \(b_n\) deviates from \(a_n\) at each position. As \(n\) becomes very large, this difference approaches 0, indicating that \(b_n\) is practically catching up to \(a_n\).

• The difference provides a measurable gap between terms of two sequences.
• If a sequence \(b_n - a_n\) converges to 0, it suggests that their corresponding terms are moving in synchrony.
• This convergence is pivotal in demonstrating that another sequence, in turn, converges to the limit of the first sequence.

Thus, by establishing that \(b_n - a_n \rightarrow 0\), we conclude that \(b_n\) must also converge to the same limit \(a\). Envision this as two runners on a track, where \(b_n\) consistently matches \(a_n\)’s pace, effectively crossing the finish line together at the limit.