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Q) Find the supremum and infimum of each of the following sets of real numbers:

(a) All numbers of the form $2^{-p} + 3^{-q} + 5^{-r}$ where $p, q$, and $r$ take on all positive integer values.

(b) $S = \{x: 3x^2 – 10x + 3 < 0 \} $ and

(c) $S = \{ x: (x-a)(x-b)(x-c)(x-d) < 0 \}$ , where $a< b<c<d$.

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Please rewrite your question so it doesn't read like an order. Please add the "homework" tag, if this is homework. And please show us how far you got, where you got stuck, what you know about the problem, etc., so we can know what kind of help you need. –  Gerry Myerson Jul 30 '12 at 12:58
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What have you tried? Where are you stuck? –  Ross Millikan Jul 30 '12 at 12:59
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1 Answer

Hints: For (a) you could think about just the $2^{-p}$ term. What is its inf and sup? You say the minimum value of $p$ is $1$ and there is no maximum.

For (b) the polynomial in the first condition is a parabola. If you solve for the points where it equals zero, the range where it is less than zero is the open interval between them.

For (c), the product of four numbers is less than zero precisely when one or three of them are less than zero and none of them is zero.

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