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Homework Statement
Problem:
Prove that if ##lim_{n→+∞} s_n## = ∞ and ##s_n## ≠ 0 for all n, then ##lim_{n→+∞} 1/s_n## = 0.
Solution:
Consider any ϵ > 0. Since ##lim_{n→+∞} s_n## = ∞, there exists some positive integer m such that, whenever n ≥ m,
##s_n## > 1/ϵ and, therefore, ####1/s_n##  0## = ##1/s_n## < ϵ. So, ##lim_{n→+∞} 1/s_n## = 0.
(I also attached the problem and its solution in the TheProblemAndSolution.jpg file.)
Homework Equations
Limits.
The Attempt at a Solution
I understand the n ≥ m part and, I suppose the ϵ > 0 and ##s_n## > 1/ϵ parts wouldn't throw me off if I understood the rest of the proof, but, I don't understand the logic behind the subtraction of 0 within the absolute value and why that is even needed because, I could just multiply both sides by ϵ/##s_n## to get 1/##s_n## < ϵ. Lastly, I don't understand how one can conclude that ##lim_{n→+∞} 1/s_n## = 0 by looking at 1/##s_n## < ϵ. To me, all that says is that ##lim_{n→+∞} 1/s_n## is less than some positive number so, it can be a smaller positive number or a negative number … this interval includes 0 but there are many other values in the interval so, I cannot see how one could simply conclude that ##lim_{n→+∞} 1/s_n## is 0.
Any help in understanding what's going on with this problem would be really appreciated!
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