To address the given problem, we need to analyze whether the statement is true or false. Specifically, we need to determine if, given a function $f: A \to \mathbb{R}$ where the limit as $x \to c$ does not exist, there necessarily exists a sequence $\{x_n\}$ in $A$ such that $x_n \to c$ but $f(x_n)$ does not converge.

Proof:

Let's assume $f: A \to \mathbb{R}$ and the limit $\lim_{x \to c} f(x)$ does not exist.

To show the desired result, we'll use the contrapositive: If for every sequence $\{x_n\}$ in $A$ such that $x_n \to c$, $f(x_n)$ converges, then the limit $\lim_{x \to c} f(x)$ exists.

1. Contrapositive Statement: Suppose for every sequence $\{x_n\} \subset A$ such that $x_n \to c$, the sequence $f(x_n)$ converges. We want to show that $\lim_{x \to c} f(x)$ exists.

2. Definition of Limit: By the definition of the limit, $\lim_{x \to c} f(x) = L$ exists if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x$ satisfying $0 < |x - c| < \delta$, we have $|f(x) - L| < \epsilon$.

3. Sequential Criterion for Limits: If $\lim_{x \to c} f(x)$ exists and equals $L$, then for every sequence $\{x_n\}$ such that $x_n \to c$, the sequence $f(x_n)$ converges to $L$.

4. Given Condition: Given that for every sequence $\{x_n\} \subset A$ such that $x_n \to c$, the sequence $f(x_n)$ converges, there exists a limit point for $f(x_n)$.

5. Uniqueness of the Limit: Assume $f(x_n)$ converges to some value $L$. If $\lim_{x \to c} f(x)$ does not exist, there must be sequences $\{x_n\}$ and $\{y_n\}$ such that $x_n \to c$ and $y_n \to c$, but $f(x_n)$ and $f(y_n)$ converge to different limits, violating the uniqueness of the limit.

6. Conclusion: Therefore, if the limit $\lim_{x \to c} f(x)$ does not exist, it must be because there exists some sequence $\{x_n\}$ such that $x_n \to c$ but $f(x_n)$ does not converge to any limit.

Thus, we have proved that if the limit $\lim_{x \to c} f(x)$ does not exist, then there exists a sequence $\{x_n\}$ such that $x_n \to c$ but $f(x_n)$ does not converge. Therefore, the statement is true.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How to prove that a function is continuous at a point?
2. What is the relationship between pointwise and uniform convergence of sequences of functions?
3. Can you explain the epsilon-delta definition of a limit?
4. How does one show that a function is uniformly continuous?
5. What are the conditions for the existence of a limit of a multivariable function?

Tip: When dealing with limits and convergence, always consider the definitions and criteria involved. It helps in structuring proofs and understanding the behavior of functions around the points of interest.