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ques 2 part b.PNG

Proof. Since xn -> infinity as n ->

infinity , as we can write lim (n -> infinity) xn = infinity.

If for every constant M >0 , there exist no in N such that xn > M for all n > no?????

ques 2 part b.PNG

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IMG_20190924_083530.jpg

Solution:
Let lim In+1 En = L. Then, for any e > 0, 3N E N such that for n > N,
Un+1-yn
- I <E=> L-E<
<L+E.
yn+1 - yn
Un+1 - Un
Using that (Un) is a strictly increasing sequence,...

IMG_20190924_083559.jpg

Sines (sin) is unbounded, we can choose 111 large enough so that ym is positive; then we may divide by gm without changing the sides of the inequalities: Sines (yfl) is unbounded, taking the limit...

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