prove or give a counterexample- if f:A->R is a function and limit x->c of f does not exist for a given c then there exists a sequence xn in A such that xn->c but f(xn) does not converge

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.

Explore the proof showing that for a function f defined on a set A to R, if the limit as x approaches c of f does not exist, then there exists a sequence x_n in A such that x_n approaches c but f(x_n) does not converge. Understand the sequential criterion for limits and the contrapositive approach used in the proof.

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