Question

A set X is said to be closed under multiplication if for every X1, X2 E X we have X1X2 E X. Let A be the union of all bounded subsets X CR that are closed under multiplication. Does inf(A) exist? If it does, find it.

Answer 1

inf(A) does not exist.

Explanation:

As per the question:

We need to prove that A is closed under multiplication,

If for every
`X_(1), X_(2)\in X`

`X_(1)X_(2)\in X`

Proof:

Suppose, x, y
`\in A`

Since, both x and y are real numbers thus xy is also a real number.

Now, consider another set B such that:

B = {xy} has only a single element 'xy' and thus [B] is bounded.

Since, [A] represents the union of all the bounded sets, therefore,

`B\subset A`

⇒ xy
`\in A`

Therefore, from x, y
`\]in A`, we have xy
`\]in A`.

Hence, set a is closed under multiplication.

Now, to prove whether inf(A) exist or not

Proof:

Let us assume that inf(A) exist and inf(A) =
`\beta`

Thus
`\beta` is also a real number.

Let C be another set such that

C = {
`\beta` - 1}

Now, we know that C is a bounded set thus {
`\beta` - 1} is also an element of A

Also, we know:

inf(A) =
`\beta`

Therefore,

`n(A)\geq \beta`

But

`\beta - 1` is an element of A and
`\beta - 1 \leq \beta`

This is contradictory, thus inf(A) does not exist.

Hence, proved.