- #1

- 818

- 8

## 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!