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:

option 3 . (5x + 1) - (-10x + 6)

step-by-step explanation:

option 1 :

(4x + 3) + (-9x + 4)

or 4x+3 -9x + 4

or -5x + 7

option 2

(x - 2) - (6x - 9)

or x-2-6x + 9

or -5x + 7

option 3

(5x+ + 6)

or 5x + 1 + 10x -6

or 15x -5

option 4

-4x+2x+7-3x

-7x + 2x + 7

-5x + 7

hence only option 3 did   not satisfy the given simplify expression.