Existence of Sequence in Set of Real Numbers whose Limit is Infimum

Theorem
Let $A \subseteq \R$ be a non-empty subset of the real numbers.

Let $b$ be an infimum of $A$.

Then there exists a sequence $\sequence {a_n}$ in $\R$ such that:
 * $(1): \quad \forall n \in \N: a_n \in A$
 * $(2): \quad \displaystyle \lim_{n \mathop \to \infty} a_n = b$

Proof
From Infimum of Subset of Real Numbers is Arbitrarily Close:

For $\epsilon = \dfrac 1 n$ there exists an $a_n \in A$ such that:
 * $a_n - b < \dfrac 1 n$

Since $b$ is an infimum of $A$:
 * $0 \le a_n - b$

Therefore:
 * $\displaystyle \lim_{n \mathop \to \infty} a_n = b$