To prove that every convergent sequence is bounded, we will use the definition of convergence and show that this property implies boundedness.

Definitions:

• Convergent Sequence: A sequence $\{a_n\}$ is said to converge to a limit $L$ if, for every $\epsilon > 0$, there exists an integer $N$ such that for all $n \geq N$, we have: $|a_n - L| < \epsilon.$ This means that as $n$ increases, the terms of the sequence get arbitrarily close to $L$.

• Bounded Sequence: A sequence $\{a_n\}$ is bounded if there exists a real number $M$ such that for all $n$, the absolute value of $a_n$ is less than or equal to $M$. In other words, there is some upper bound $M$ such that: $|a_n| \leq M \quad \text{for all} \, n.$

Proof:

Let $\{a_n\}$ be a convergent sequence, and let it converge to a limit $L$. By the definition of convergence, for every $\epsilon > 0$, there exists an integer $N$ such that for all $n \geq N$, we have: $|a_n - L| < \epsilon.$

Let's choose $\epsilon = 1$. This means that there exists an integer $N$ such that for all $n \geq N$, we have: $|a_n - L| < 1.$ This implies that: $|a_n| < |L| + 1 \quad \text{for all} \, n \geq N.$

Now, consider the first $N-1$ terms of the sequence $\{a_n\}$, namely $a_1, a_2, \dots, a_{N-1}$. These terms are finite in number, and therefore, they are bounded. Let $M_1$ be an upper bound for these first $N-1$ terms, i.e., there exists a constant $M_1$ such that: $|a_n| \leq M_1 \quad \text{for} \, n = 1, 2, \dots, N-1.$

Next, let $M_2 = |L| + 1$, which bounds the terms $a_n$ for $n \geq N$, i.e.: $|a_n| \leq M_2 \quad \text{for} \, n \geq N.$

Thus, the sequence $\{a_n\}$ is bounded by the maximum of $M_1$ and $M_2$. Let $M = \max(M_1, M_2)$. Therefore, for all $n$, we have: $|a_n| \leq M.$

Hence, the sequence $\{a_n\}$ is bounded.

Conclusion:

We have shown that if $\{a_n\}$ is a convergent sequence, then there exists a real number $M$ such that $|a_n| \leq M$ for all $n$. Therefore, every convergent sequence is bounded.

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Would you like more details on this proof or any clarifications?

Here are 5 related questions for further exploration:

1. What happens if a sequence diverges? Is it always unbounded?
2. How do we formally define the limit of a sequence?
3. Can a bounded sequence still fail to converge?
4. What is the difference between a convergent sequence and a Cauchy sequence?
5. How do we use the $\epsilon-\delta$ definition in proving convergence?

Tip: Whenever dealing with limits and sequences, visualizing a sequence on a number line can often help in understanding how it behaves as $n$ increases.