Question

If a ∈ R is such that o , ≤ a ≤ ε ∀ ε > 0 ,then show that a = 0.

Answer 1

Answer on question #69116 – Math – Real Analysis

Question

If $a \in R$ is such that $0 \leq a \leq \varepsilon \ \forall \varepsilon > 0$, then show that $a = 0$.

Solution

Let $a \neq 0$ and $a = \delta > 0$, then there exists some $\varepsilon: \delta > \varepsilon > 0$ for which the condition $0 \leq a \leq \varepsilon$ does not hold.

Hence the assumption $a \neq 0$ is not true, so $a = 0$.

Answer: $a = 0$.

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