Proof: A Convergent Sequence is Bounded

Let $(a_n)$ be a sequence that converges to a limit $L$. By the definition of convergence:

For every $\epsilon > 0$, there exists a positive integer $N$ such that for all $n \geq N$, $|a_n - L| < \epsilon.$

Step 1: Establish bounds for terms with $n \geq N$

Let $\epsilon = 1$. By the definition of convergence, there exists $N$ such that for all $n \geq N$, $|a_n - L| < 1.$ Rewriting this inequality, $-1 < a_n - L < 1 \quad \implies \quad L - 1 < a_n < L + 1 \quad \text{for all } n \geq N.$ Thus, the terms $a_n$ for $n \geq N$ are bounded between $L - 1$ and $L + 1$.

Step 2: Bound the terms with $n < N$

The sequence $(a_n)$ has a finite number of terms $a_1, a_2, \dots, a_{N-1}$. Let $M = \max \{|a_1|, |a_2|, \dots, |a_{N-1}|, |L| + 1\}.$

Step 3: Combine the bounds

Now, for all $n \in \mathbb{N}$,

• If $n \geq N$, $|a_n| \leq |L| + 1 \leq M$,
• If $n < N$, $|a_n| \leq M$ by the definition of $M$.

Thus, $|a_n| \leq M$ for all $n \in \mathbb{N}$, which proves that the sequence $(a_n)$ is bounded.

Conclusion

Every convergent sequence is bounded.

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Related Questions

1. What is the definition of a bounded sequence?
2. Can a bounded sequence diverge? Provide an example.
3. Does the converse hold: Is every bounded sequence convergent?
4. How does this proof extend to sequences in $\mathbb{R}^n$?
5. How does the concept of boundedness relate to the Bolzano-Weierstrass theorem?

Tip: Understanding how convergence implies boundedness can help when analyzing limits of sequences in calculus or real analysis!