I need assistance with #2 please. Thank you! According to Theorem 1.19, there ex

Question

I need assistance with #2 please.

Thank you!

According to Theorem 1.19, there exists an order field R which has the lease upper bound property and Q E R. (1) Now, let A1+EN SR, by above, show that Sup(A)e exists. (Hint: (1 + 1/n)"-1 + (r) (1/n) + (3) (1/n)2 + + (1/n)") (2) Following (1), suppose that instead of least upper bound property, R is equipped with Greatest lower bound property only (that is, for any nonempty, bounded from below set in R, there exist a greatest lower bound) Show that the least upper bound of A still exist

Explanation / Answer

let the set of real numbers has the greatest lower bound property i.e. for any nonempty set bounded from below, in R has a greatest lower bound.

then let A be a nonempty subset of R and bounded below, let a is glb of A, then x>=a for all x in A.

Now let us consider the set -A subset of R then ofcourse this set is bounded above by -a as x<=-a for all x in -A. so that -A is a subset of R which is bounded above and has a least upper bound namely -a, i.e. additive inverse of glb of A so that R has lub property.

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