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Let A subset or equal of R be bounded above and let c belong to R.Define the sets c+A and cA by c+A={c+a : a belong to A} and cA={ca : a belong to A}.
1-show that sup(c+A)=c+ Sup A
2-if c>=0,show that sup(cA)=cSupA
3-postulate a similar type of statement for sup(cA)for the case c<0.

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Solution Summary

This solution is a proof regarding bounded subsets.

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Proof:
1. For any a in A, c+a<=sup(c+A). Let b=supA, we can find a sequence ai in A such that lim(an)(as n->inf)=b. Since c+an<=sup(c+A), then let n->inf, we get ...

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