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## If the limit is greater than a number, so are all but finitely many of numbers in this sequence

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.10 This part actually has been shown in Exercise 8.9. Solution: Suppose $\lim s_n=s$. Then we have…

## If the sequence is greater than a number, so is the limit

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.9 Solution: Part aWe argue it by contradiction. Suppose $\lim s_n=s < a$. Then $a-s >0$. Since…

## Prove the limits equal to the corresponding numbers

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.8 We shall use the following useful formula. $\sqrt{a}-b=\dfrac{a-b^2}{\sqrt{a}+b}$, which is a variation of $(a-b)(a+b)=a^2-b^2$. Solution: Part…

## Show the limit does not exist

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.7 Solution: Part aSuppose the limit exists and equals $a$. Then for $\epsilon=\dfrac{1}{2}$, there exists $N>0$ such…

## When the limit of a sequence is zero

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.6 Solution: Part aWe first show that if $\lim s_n=0$ then $\lim |s_n|=0$. Let $\epsilon>0$. Since \$\lim…