a and b are bounded non-empty subsets of r. for inf(a) to be less than or equal to inf(b), which of the following conditions must be met?
a) for every b in b and epsilon > 0, there exists a in a, such that a < b + epsilon.
b) there exists a in a, and b in b such that a < b.
if neither of these conditions are appropriate, what would be appropriate conditions for inf(a) to be less than or equal to inf(b)?

a) must be met

Step-by-step explanation:

We have two conditions:

a) For every and , there exists , such that .

b) There exists and such that  .

We will prove that conditon a) is equivalent to

If a) is not satisfied, then it would exist and such that, for every , . This implies that is a lower bound for A and in consequence

Then, implies a).

If is not satisfied then, and in consequence exists such that . Then and, for every ,

.

So, a) is not satisfied.

In conclusion, a) is equivalent to

Finally, observe that condition b) is not an appropiate condition to determine if or not. For example:

n=9

step-by-step explanation:

7 times a number = 7n

decreased by 13 = 7n-13

same as is =

5 times the number = 5n

increased by 5 = 5n+5

7n-13 = 5n+5

2n=18

n=9

answer: 204

step-by-step explanation:

we are told that the revenue is

x⋅(4080−10x)=−10x^2+4080x

this is a quadratic function, so to maximize it we should find the x-value of its vertex. recall that the x-value of the vertex of the parabola ax^2+bx+c is given by the formula −b/2a.

so in this case, we have a=−10 and b=4080, so the x-value of the vertex is

−b/2a= −(4080)/2(−10)=204