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:

$(1): \quad \forall n \in \N: a_n \in A$

$(2): \quad \ds \lim_{n \mathop \to \infty} a_n = b$

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:

$\ds \lim_{n \mathop \to \infty} a_n = b$

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Corollary $5.7$