Let (x_n) and (y_n) be sequences of real numbers such that x_n >= 0 and y_n >= 0

Question

Let (x_n) and (y_n) be sequences of real numbers such that x_n >= 0 and y_n >= 0 for n is an element of N. Prove that the lim sup (x_n)(y_n) <= (lim sup(x_n))(lim sup(y_n)).....all three have n going to infinity. Provide that the product on the right is not of the form 0*infinity. Demonstrate that the inequality can be strict. Please provide proof.

Explanation / Answer

Because L > (L+1)/2 > 1, there exists an M such that when n>M, xn+1 / xn > (L+1)/2, and thus xn+1 > xn*(L+1)/2. Suppose an upper bound U exists. By induction, for n>M, we have xn+k > ((L+1)/2)^k * xn. U is the upper bound, so U >= ((L+1)/2)^k * xn must be true. But this means log(U) >= k*log((L+1)/2) + log(xn), then log(U) - log(xn) >= k*log((L+1)/2), then (log(U) - log(xn)) / (log((L+1)/2)) >= k. This is a contradiction because k --> infinity while (log(U) - log(xn)) / (log((L+1)/2)) is constant. Thus there is no upper bound U.

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