Limit of Function by Convergent Sequences/Corollary

Theorem
Let $\left({a \,.\,.\, b}\right)$ be an open real interval.

Let $f: \left({a \,.\,.\, b}\right) \to \R$ be a real function.

Let $l \in \R$.

Then:
 * $(1): \quad \displaystyle \lim_{x \mathop \to a^+} f \left({x}\right) = l \iff \forall \left \langle {x_n} \right \rangle \subseteq \left({a \,.\,.\, b}\right): \lim_{n \mathop \to \infty} x_n = a \implies \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$
 * $(2): \quad \displaystyle \lim_{x \mathop \to b^-} f \left({x}\right) = l \iff \forall \left \langle {x_n} \right \rangle \subseteq \left({a \,.\,.\, b}\right): \lim_{n \mathop \to \infty} x_n = b \implies \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$

where:
 * $\displaystyle \lim_{x \mathop \to a^+} f \left({x}\right)$ denotes the limit of $f$ from the right
 * $\displaystyle \lim_{x \mathop \to b^-} f \left({x}\right)$ denotes the limit of $f$ from the left.

Proof
We have that $\left({\R, d}\right)$ is a metric space, where $d$ is the Euclidean metric on $\R$.

Thus the theorem follows immediately from Limit of Function by Convergent Sequences.