Limit of Function by Convergent Sequences/Corollary

Theorem
Let $\openint a b$ be an open real interval.

Let $f: \openint a b \to \R$ be a real function.

Let $l \in \R$.

Then:
 * $(1): \quad \ds \lim_{x \mathop \to a^+} \map f x = l \iff \forall \sequence {x_n} \subseteq \openint a b: \lim_{n \mathop \to \infty} x_n = a \implies \lim_{n \mathop \to \infty} \map f {x_n} = l$
 * $(2): \quad \ds \lim_{x \mathop \to b^-} \map f x = l \iff \forall \sequence {x_n} \subseteq \openint a b: \lim_{n \mathop \to \infty} x_n = b \implies \lim_{n \mathop \to \infty} \map f {x_n} = l$

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

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

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