# Distance from Subset to Infimum

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## Theorem

Let $S \subseteq \R$ be a subset of the real numbers.

Suppose that the infimum $\inf S$ of $S$ exists.

Then:

- $\map d {\inf S, S} = 0$

where $\map d {\inf S, S}$ is the distance between $\inf S$ and $S$.

## Proof

By Distance between Element and Subset is Nonnegative:

- $\map d {\inf S, S} \ge 0$

By definition of infimum:

- $\forall \epsilon > 0: \exists s \in S: \map d {\inf S, s} < \epsilon$

meaning that, by nature of the infimum and the definition of $\map d {\inf S, S}$:

- $\forall \epsilon > 0: \map d {\inf S, S} < \epsilon$

Together, these two observations lead to the conclusion that:

- $\map d {\inf S, S} = 0$

as desired.

$\blacksquare$