Theorem: When it exists, the limit of a sequence is unique.

Proof:

Let $\{a_n\}$ be a sequence, and suppose that this sequence converges to two distinct limits. Specifically, let:

$\lim_{n \to \infty} a_n = L_1 \quad \text{and} \quad \lim_{n \to \infty} a_n = L_2.$

We aim to show that $L_1 = L_2$, thus proving that the limit, if it exists, is unique.

Step 1: Definition of Limit

By the definition of a limit of a sequence, we know:

1. Since $\lim_{n \to \infty} a_n = L_1$, for every $\epsilon > 0$, there exists an integer $N_1$ such that for all $n \geq N_1$, we have:

$|a_n - L_1| < \epsilon.$

2. Similarly, since $\lim_{n \to \infty} a_n = L_2$, for every $\epsilon > 0$, there exists an integer $N_2$ such that for all $n \geq N_2$, we have:

$|a_n - L_2| < \epsilon.$

Step 2: Choose an Arbitrary $\epsilon > 0$

Now, let's consider an arbitrary positive number $\epsilon > 0$. We need to show that $L_1 = L_2$.

Step 3: Choose $N = \max(N_1, N_2)$

Since both $N_1$ and $N_2$ exist, we can choose $N = \max(N_1, N_2)$, which means that for all $n \geq N$:

$|a_n - L_1| < \epsilon \quad \text{and} \quad |a_n - L_2| < \epsilon.$

Step 4: Use the Triangle Inequality

Now, consider the absolute difference between $L_1$ and $L_2$:

$|L_1 - L_2| = |(L_1 - a_n) + (a_n - L_2)|.$

By the triangle inequality, we have:

$|L_1 - L_2| \leq |L_1 - a_n| + |a_n - L_2|.$

For $n \geq N$, we know that both $|L_1 - a_n| < \epsilon$ and $|a_n - L_2| < \epsilon$. Therefore:

$|L_1 - L_2| \leq \epsilon + \epsilon = 2\epsilon.$

Step 5: Conclusion

Since $\epsilon$ is arbitrary, we can make it as small as we like. This implies that:

$|L_1 - L_2| \leq 2\epsilon \quad \text{for every} \quad \epsilon > 0.$

Thus, $L_1 = L_2$, which shows that the limit of a sequence, when it exists, is unique.

Final Conclusion:

We have proven that the limit of a sequence, when it exists, must be unique.

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Would you like any more details or clarifications? Here are some related questions you might find useful:

1. What is the formal definition of the limit of a sequence?
2. Can a sequence have more than one limit? Under what conditions?
3. How do the concepts of convergence and boundedness relate in sequences?
4. What happens if a sequence does not converge?
5. What is the difference between a limit point and a limit of a sequence?

Tip: Always remember that in the context of sequences, the limit is only defined if the terms of the sequence get arbitrarily close to a specific value as $n \to \infty$. If they oscillate or fail to approach any particular value, the limit doesn't exist.