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I am new to proofs and am still struggling to parse them. I am not looking for a proof to the following statement; just guidance as to where to start or what the shape of a proof for it looks like.

Prove that a subset $A$ of $\mathbb{R}$ is bounded if and only if there is $M ∈ \mathbb{R}$ such that $|x| ≤ M$ for all $x ∈ A$.

I know something is bounded if it has an upper and lower bound, and I have read axioms of fields and the definition of the absolute value.

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    $\begingroup$ That looks like the definition of a bounded subset of $\Bbb R$ to me. So, presumably you have a different definition of bounded set that you are working from. Surely, the first thing to do is to write down that definition, and then check to see whether a set satisfying the given criterion satisfies it. $\endgroup$ – Lord Shark the Unknown Apr 12 at 4:36
  • $\begingroup$ thank you, that makes a lot of sense, but there is no other definition. the text above is all that was provided. $\endgroup$ – tau Apr 12 at 5:42
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    $\begingroup$ @LordSharktheUnknown: another definition might be like this: a non-empty set $S$ is bounded if there exist numbers $m, M$ such that $m\leq s\leq M$ for all $s\in S$. Your definition is simpler and my preferred one. $\endgroup$ – Paramanand Singh Apr 12 at 15:43

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