Anyone can help me out? Limit of a function at an accumulation point.

[monkey372]

1. Homework Statement
Let f : D → R and c ∈ R an accumulation point of D.

2. Homework Equations

Prove the following are equivalent:

(a) f has a limit at c.

(b) For all sequences (sn ) such that c does not equal to sn ∈ D for all n ∈ N and sn → c,
the sequence (f (sn )) is convergent in R.3. The Attempt at a Solution
Here is my approach:
Let x be an arbitrary element of D. Since f has a limit at c which means f(x) -> L as x -> c, we have the sequence xn -> c, where (xn) does not equal to c for all natural number n, as f(xn)->c.

But then I don't know how I should continue. Please help me out.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[lippyka]

We will prove (a) implies (b). Suppose f has a limit at c, then for all sequences (sn) such that c does not equal to sn ∈ D for all n ∈ N and sn → c, we have that lim f(sn) = lim f(x) as x -> c. Since the sequence (sn) converges to c, it follows that the sequence (f(sn)) converges to the same limit, namely f(c). Thus, (b) is true.The proof of (b) implies (a) follows similarly. If for all sequences (sn) such that c does not equal to sn ∈ D for all n ∈ N and sn → c, we have that the sequence (f(sn)) converges to some limit L, then it follows that f(x) -> L as x -> c. Thus, (a) is true.