Limit of Function by Convergent Sequences/Real Number Line

Theorem
Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$, except possibly at the point $c \in \left({a \,.\,.\, b}\right)$.

Then $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$ iff:
 * for each sequence $\left \langle {x_n} \right \rangle$ of points of $\left({a \,.\,.\, b}\right)$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\displaystyle \lim_{n \to \mathop \infty} x_n = c$

it is true that:
 * $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$

Necessary Condition
Let $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$.

Let $\epsilon > 0$.

Then by the definition of the limit of a function:
 * $\exists \delta > 0: \left|{f \left({x}\right) - l}\right| < \epsilon$

provided $0 < \left|{x - c}\right| < \delta$.

Now suppose that $\left \langle {x_n} \right \rangle$ is a sequence of points of $\left({a \,.\,.\, b}\right)$ such that:
 * $\forall n \in \N_{>0}: x_n \ne c$

and:
 * $\displaystyle \lim_{n \to \infty} x_n = c$

Since $\delta > 0$, from the definition of the limit of a function:
 * $\exists N: \forall n > N: \left|{x_n - c}\right| < \delta$

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

That means:
 * $0 < \left|{x_n - c}\right| < \delta$

But that implies:
 * $\left|{f \left({x_n}\right) - l}\right| < \epsilon$

That is, given a value of $\epsilon > 0$, we have found a value of $N$ such that:
 * $\forall n > N: \left|{f \left({x_n}\right) - l}\right| < \epsilon$

Thus:
 * $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$

Sufficient Condition
Suppose that for each sequence $\left \langle {x_n} \right \rangle$ of points of $\left({a \,.\,.\, b}\right)$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\displaystyle \lim_{n \mathop \to \infty} x_n = c$, it is true that:
 * $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$

Aiming for a contradiction, suppose it is not true that:
 * $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$

Thus:
 * $\exists \epsilon > 0: \forall \delta > 0: \exists x: 0 < \left|{x - c}\right| < \delta: \left|{f \left({x_n}\right) - l}\right| \ge \epsilon$

In particular, if $\delta = \dfrac 1 n$, we can find an $x_n$ where $0 < \left|{x - c}\right| < \dfrac 1 n$ such that:
 * $\left|{f \left({x_n}\right) - l}\right| \ge \epsilon$

But then $\left \langle {x_n} \right \rangle$ is a sequence of points of $\left({a \,.\,.\, b}\right)$ such that:
 * $\displaystyle \forall n \in \N_{>0}: x_n \ne c$ and $\lim_{n \mathop \to \infty} x_n = c$

but for which it is not true that:
 * $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$

The result follows from this contradiction