Let (xn) be an unbounded sequence and (yn) be a convergent sequence. Prove that (xn+yn) is an unbounded sequence.
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Let $sum X_n$ and $sum Y_n$ be two series of real numbers. Suppose $X_n eq 0$ for all $n geq 1$ and $sum X_n$ converges absolutely. Suppose further that $lim(Y_n/X_n)$ exists as $n$ approaches infinity. Prove that both $sum Y_n$ and $sum Y_n^2$ converge. (Hint: The sequence ${Y_n/X_n}$ is convergent, so it is bounded.)
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