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monkey372

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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.

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

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