"assume that s is a non-empty set of real numbers which is bounded above and lambda is the least upper bound. prove that for all real numbers beta is less than lambda there is a real number alpha in s such that beta is less than alpha"

By definition, if is the least upper bound of the set , it means two thing:

In other words, the least upper bound of a set is greater than or equal to every single element of the set, but it is "close enough" to the elements of the set, because you guaranteed to find elements in the set between and

For example, pick . Obvisouly, the least upper bound is . In fact, every number in is smaller than 10, but as soon as you take away something from 10, say 0.01, you get 9.99, and there are elements in greater than 9.99, say 9.9999.

So, the claim is basically proven by definition: if , let . By definition, there exists .

s

step-by-step explanation:

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .