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4.3.10_Study_Project_Extending_the_algebra_of_limits[1]

# By the definition of tending to

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Unformatted text preview: _ . Therefore, xn > 4 for all n ≥ K1 . Since lim(yn ) = ∞ , there exists K2 ∈ such that for all n ≥ K2 , _______ > __________ . Let K = ___________ . Then for n ≥ K it follows that xn + yn > 4 + ______ ______ M . Therefore, by the definition of tending to infinity, lim( xn + yn ) = ∞ . c) Suppose ( xn ) converges and lim(yn ) = ∞ . Modify the proof above, without splitting it into separate cases, to show that lim( xn + yn ) = ∞ . 3. Formalizing the observations Using your results in Parts 1 and 2, extend the theorem on algebra of limits to draw conclusions about products and sums of sequences, where one sequence converges and the other tends to infinity. In your extension, be sure to clearly state the hypothesis and the conclusion. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE ­0837810...
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