Let $p$ be a prime number and a set $A = \left\{-\dfrac{m} {n} - p \dfrac{n} {m} : m, n \in \mathbb{N} \right\}$ Evaluate $\sup (A)$.

My attempt:

$\displaystyle \sup (A)=-\inf({m\over n}+{{pn}\over m}) \\ =-\inf_{{t\in {\bf Q}^+}} (t+{p\over t})=-2\sqrt{p}$

I have a trouble in $p$. Thinking that it is irrelevant to be a prime number, but it should be $p > 0$.

Is my approachment fine?

  • $\begingroup$ Yes, it's fine. $\endgroup$ – Paul Frost Jun 3 '18 at 17:31

Your answer is fine. There's a sequence of Q, say (an), that approaches p^(1/2). then A = {x = an + p/an} has: inf(A) = 2 * p^(1/2), due to the continuity of the function f(x) = x + p/x.


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