Every Convergent Sequence is Bounded: A Comprehensive Analysis and Its Converse

Sequences are a fundamental concept in mathematics, particularly in analysis, with numerous properties and behaviors that are of interest. One of the most significant properties of a convergent sequence is that it is bounded. In this article, we will explore the proof that every convergent sequence is bounded and then delve into the question of whether the converse is true. We will also examine a partial converse and provide examples to illustrate these concepts.

Proving That Every Convergent Sequence is Bounded

A sequence $(a_n)$ is said to converge to a limit $ell$ if for every $varepsilon>0$, there exists a positive integer $N$ such that $|a_n - ell|

Proof that Convergent Sequences are Bounded

Consider a convergent sequence $(a_n)$. By definition, there exists a limit $ell in mathbb{R}$ such that for all $varepsilon > 0$, there is a positive integer $N$ for which $|a_n - ell| [ ell - 1 Let $M max{lvert a_1 rvert, lvert a_2 rvert, ldots, lvert a_{N-1} rvert, lvert ell - 1 rvert, lvert ell 1 rvert}$. For $n geq N$, we have:

Example of a Non-convergent Bounded Sequence

A standard example of a bounded sequence that is not convergent is the alternating sequence $1, -1, 1, -1, ldots$. This sequence is bounded because $|a_n| leq 1$ for all $n$, but it does not converge as it oscillates and does not approach a single limit.

Converse: Is Every Bounded Sequence Convergent?

The converse of the statement "every convergent sequence is bounded" is not true. In other words, it is not necessary that every bounded sequence is convergent. To understand why, consider the sequence $(-1)^n$. This sequence is bounded because each term is either $-1$ or $1$, but it does not converge because it oscillates between these two values and does not approach a single limit.

Partial Converse: Bounded Sequences and Subsequences

Even though every convergent sequence is bounded, there is a partial converse that is true in the context of the real numbers. In the reals, any bounded sequence has a subsequence that converges. This is a fundamental result in real analysis known as the Bolzano-Weierstrass theorem, which states that every bounded sequence in $mathbb{R}$ has a convergent subsequence.

Conclusion

In conclusion, every convergent sequence is indeed bounded, as demonstrated by the rigorous proof provided. However, the converse is not true, and there exist bounded sequences that are not convergent. Nonetheless, a bounded sequence in the real numbers will always have a convergent subsequence, which is a powerful and useful result in mathematical analysis.

Note: Understanding the distinction between convergent and bounded sequences is crucial for deeper insights into the nature of sequences and their convergence properties in real and complex analysis.