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Unformatted text preview: not hold? 1 Problem. Let ( x n ) and ( y n ) be bounded sequences of nonnegative real numbers. Let x * = lim sup ( x n ). Prove that if y * = lim ( y n ) exists, then lim sup ( x n y n ) = x * y * . Solution Set z * = lim sup n x n y n . Choose > 0. Then there exists an N > 0 such that n > N = y * < y n < y * + . Therefore, if we set u m = sup n m x n , v m = sup n m x n y n , then for m > N we have v m = sup n m x n y n ( y * ) sup n m x n = ( y * ) u m . Hence z * = lim m v m ( y * ) lim m u m = ( y * ) x * . Since is arbitrary, we conclude that z * y * x * . Also, the preceding problem implies that z * x * y * (why?), so we conclude that we have z * = x * y * . 2...
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 Summer '07
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 Math

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