The world of sequences in mathematics can be fascinating, filled with patterns and behaviors that sometimes defy our initial intuition. A particularly interesting question arises when we consider sequences that are “bounded” – meaning they stay within a certain range. This leads us to the central question: Is Every Bounded Sequence Convergent Sequence? The short answer, as we’ll explore, is no. While boundedness is a necessary condition for convergence, it is not sufficient. There exist bounded sequences that stubbornly refuse to settle down to a single limiting value.
Unpacking Boundedness and Convergence
To truly understand why “Is Every Bounded Sequence Convergent Sequence” is false, let’s break down the concepts involved. A sequence is considered bounded if all its terms lie within a finite interval. In other words, there’s a number M such that the absolute value of every term in the sequence is less than or equal to M. Think of it as the sequence being trapped between two walls, never able to escape. For example, the sequence {sin(n)} is bounded between -1 and 1. This is a necessary ingredient for convergence, as a sequence cannot converge if its terms are flying off to infinity.
On the other hand, convergence means that the terms of the sequence get arbitrarily close to a specific number, called the limit, as we move further and further along the sequence. Formally, for any small positive number (epsilon), we can find a point in the sequence such that all subsequent terms are within epsilon of the limit. The critical distinction is that while a convergent sequence must be bounded, a bounded sequence doesn’t necessarily have to settle down to a single value. Consider these different scenarios:
- Convergent and Bounded: {1/n} converges to 0 and is bounded between 0 and 1.
- Divergent and Unbounded: {n} diverges to infinity and is unbounded.
- Divergent but Bounded: This is the tricky one, and the key to answering our question!
A classic example of a bounded but divergent sequence is {(-1)^n}. This sequence alternates between -1 and 1. It’s bounded because all its terms are either -1 or 1. However, it’s divergent because it doesn’t approach a single limit. It oscillates endlessly. We can represent this more clearly in a table:
| n | (-1)^n |
|---|---|
| 1 | -1 |
| 2 | 1 |
| 3 | -1 |
| 4 | 1 |
As you can see, the sequence keeps “jumping” around, never settling close to one particular value. This highlights the fact that boundedness only restricts the *range* of the sequence’s values; it doesn’t guarantee that the sequence will actually *approach* a limit. The oscillation prevents convergence.
To delve deeper into the nuances of sequences and their convergence properties, and to solidify your understanding with more examples and rigorous proofs, consider exploring a more in-depth treatment of the subject. This deeper understanding will help you to master the material.