Exercise 1.2.10

Question

Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not: \begin{enumerate}[label=(\alph*)] \item Two real numbers satisfy $a0$. \item Two real numbers satisfy $a0$. \item Two real numbers satisfy $a \leq b$ if and only if $a0$. \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] \item False, if $a=b$ then $a < b + \epsilon$ for all $\epsilon > 0$ but $a ot < b$ \item False, consider $a=b$ as above \item True. First suppose $a < b + \epsilon$ for all $\epsilon > 0$, We want to show this implies $a \le b$. We either have $a \le b$ or $a > b$, but $a > b$ is impossible since the gap implies there exists an $\epsilon$ small enough such that $a > b + \epsilon$. Second suppose $a \le b$, obviously $a < b + \epsilon$ for all $\epsilon > 0$. \end{enumerate}

Exercise 1.2.10

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Exercise 1.2.10

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