Real Analysis Proof (Limits of Functions)

[jmjlt88]

Let A be a subset of ℝ. Let c be a limit point of A. Consider the function f: A → ℝ

Claim: If the function f has not have a limit at c, then there exists a sequence (x_n), where x_n≠c for all n, such that lim x_n=c, but the sequence (f(x_n)) does not converge.

Since the function f does not have a limit at c, for all L in ℝ, there is some ε_L>0 such that for all δ>0 there is a point x in A satisfying 0<|x-c|<δ and yet |f(x)-L|≥ε_L.
I know how to create a sequence that converges to 0; I am having trouble making the sequence (f(x_n)) not converge.

My initial idea was to pick points x_n such that 0<|x_n-c|<1/n and |f(x_n)-n|≥ε_n for all n. However, this was not quite working out. Any bump in the right direction would be awesome!

 

[verty]

Hint: the value f(c), if defined, is irrelevant.

I'm trying to hint without being obvious.

 

[pasmith]

Is f assumed to be continuous except at c?

In what ways can a continuous function on an open interval fail to have a limit at an endpoint?

 

[jmjlt88]

To pasmith, no, f is not assumed to be continuous.

To verty, indeed. A couple ways the sequence (f(x_n)) could fail to converge is if we have some asymptotic behavior or something like the signum function. Unfortunately, the hint provided is not getting the wheels in motion. :(

 

[verty]

Think about the contrapositive of the claim, it may be easier to prove.

 

[jmjlt88]

The contrapositive of the claim would be:
"If for every sequence (x_n) that converges to c, the sequence (f(x_n)) converges, then f has a limit at c."

I tried the contrapositive (and also had some difficulty). However, I gave up rather quickly since I am so focus on constructing that sequence to prove the original claim. I would really like to move on (as I am only reviewing the major topics from introductory analysis). As such, another hint concerning the original claim would be fantastic!

 

[pasmith]

The contrapositive of the claim would be:
"If for every sequence (x_n) that converges to c, the sequence (f(x_n)) converges, then f has a limit at c."

I tried the contrapositive (and also had some difficulty).

How far did you get?

Did you show that the limit of the sequence (f(x_n)) is independent of the sequence (x_n)?