Limit of Function by Convergent Sequences/Real Number Line

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Theorem

Let $f$ be a real function defined on an open interval $\openint a b$, except possibly at the point $c \in \openint a b$.


Then $\ds \lim_{x \mathop \to c} \map f x = l$ if and only if:

for each sequence $\sequence {x_n}$ of points of $\openint a b$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\ds \lim_{n \mathop \to \infty} x_n = c$
it is true that:
$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$


Proof

Necessary Condition

Let $\ds \lim_{x \mathop \to c} \map f x = l$.

Let $\epsilon \in \R_{>0}$.

Then by the definition of the limit of a real function:

$\exists \delta \in \R_{>0}: \size {\map f x - l} < \epsilon$

provided $0 < \size {x - c} < \delta$.

Now suppose that $\sequence {x_n}$ is a sequence of elements of $\openint a b$ such that:

$\forall n \in \N_{>0}: x_n \ne c$

and:

$\ds \lim_{n \mathop \to \infty} x_n = c$

Since $\delta > 0$, from the definition of the limit of a real function:

$\exists N \in \R_{>0}: \forall n > N: \size {x_n - c} < \delta$

But:

$\forall n \in \N_{>0}: x_n \ne c$

That means:

$0 < \size {x_n - c} < \delta$

But that implies:

$\size {\map f {x_n} - l} < \epsilon$

That is, given a value of $\epsilon > 0$, we have found a value of $N$ such that:

$\forall n > N: \size {\map f {x_n} - l} < \epsilon$

Thus:

$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$

$\Box$


Sufficient Condition

Suppose that for each sequence $\sequence {x_n}$ of elements of $\openint a b$ such that:

$\forall n \in \N_{>0}: x_n \ne c$ and $\ds \lim_{n \mathop \to \infty} x_n = c$, it is true that:
$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$


Aiming for a contradiction, suppose it is not true that:

$\ds \lim_{x \mathop \to c} \map f x = l$

Thus:

$\exists \epsilon \in \R_{>0}: \forall \delta \in \R_{>0}: \exists x \in \openint a b: 0 < \size {x - c} < \delta: \size {\map f {x_n} - l} \ge \epsilon$

In particular, if $\delta = \dfrac 1 n$, we can find an $x_n$ where $0 < \size {x - c} < \dfrac 1 n$ such that:

$\size {\map f {x_n} - l} \ge \epsilon$

But then $\sequence {x_n}$ is a sequence of elements of $\openint a b$ such that:

$\forall n \in \N_{>0}: x_n \ne c$ and $\ds \lim_{n \mathop \to \infty} x_n = c$

but for which it is not true that:

$\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$

The result follows by Proof by Contradiction.

$\blacksquare$


Sources

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