Implication of infimum

Question

Let $R = \inf\{ r: S(r) \text{ is true} \}$, then can I say $S(R+\epsilon)$ is true for every $\epsilon > 0$ ?

If no, then are there more constraints/qualifications which make this true?

If yes, then is it axiomatic/by definition or does it require some proof?

EDIT: some added constraints

1. If $S(r_0)$ is true, then $S(r)$ is true $\forall $ $r \geq r_0$
2. $\{r: S(r) \text{ is true} \}$ is nonempty

I think given these, the very first statement is true ... ?

Answer 1

We can say that for any $\epsilon>0$ there exists $r$ such that $S(r)$ holds and $R\leq r\leq R+\epsilon$. _(@1)

(This is because if such $r$ doesn't exist then $R+\epsilon$ will be new lower bound ,contradictory to definition of $R$.)

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Now suppose for some $\epsilon _0>0$ , $S(R+\epsilon_0)$ is not true ,but from (@1),there must exist $r$ such that $S(r)$ is true and $R\leq r \leq R+\epsilon_0$ ,but then this would contradict your condition 1.