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    1) Let > 0 and >0, and . Show that and are - neighborhoods of for appropriate values of .

    2)Let and be nonempty sets and let : have bounded range in . Let :
    and be defined by


    Prove that

    We sometimes express this by writing


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    Solution Preview

    Please see the attachment.

    Problem #1
    As a math convention, is an -neighborhood of . Similarly, is a -neighborhood of .
    If , ...

    Solution Summary

    This is a proof regarding nonempty sets. The nonempty set bounded range is examined.