Re: Finding sups and infs


vsgdp wrote:
An exercise asks me to find the sup and inf of some sets. For example, the
set S = {x : x = 2^(-p) + 3^(-q) + 5^(-r), where p, q, r positive integers}.
Each term is decreasing strictly, so the sup is just given when p = q = r =
1. The inf is clearly zero also since the terms are strictly decreasing
arbitrarily close to 0 (but do I need to be more rigorous than this, if so,
how without the definition of limit being introduced yet).

Also I'm stumped on this one: S = {x: (x-a)(x-b)(x-c)(x-d) < 0, a < b < c <
d}. This is some 4th degree polynomial with roots at a, b, c, and d.
Obviously the sup is 0. But how do I find the inf?

a, b, c and d divide up the real numbers into separate intervals:
[-inf,a), [a,b), [b,c), [d,inf].

(It might be more sensible to divide it up slightly differently, i.e.
[-inf,a], (a,b], etc.... )

As you move from one region to another, the signs of the terms
like (x-a), (x-b) change. For instance, for x below a, x is also below
b, c and d. Hence all four terms are negative and their product
is positive.

Does that give you enough of a hint?

- Randy

.

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