Prove that lim gn for n→∞ exists, and find it.

[dannysaf]

Suppose g1 , g2 ,... are any numbers that satisfy the inequalities
0 < gn < 1 and (1 − gn)gn+1 > 1/4 for all n.

Prove that lim gn for n→∞ exists, and find it.


I need well substantiated answer! Thanks.

 

[VietDao29]

I need well substantiated answer! Thanks.

Since you are pretty new to the forum, I'll explain some of the rules for you to understand how we operate. :)

If you're really looking for some "well substantiated answer" here, then, I'm very sorry to inform you that you've come to the wrong place. :( Don't miss the https://www.physicsforums.com/showthread.php?t=94383" that lie on top of every Homework Helping board. :)

We are here to guide you to tackle some problem, or to help you understand some concepts you find hard, and impossible to grasp. We, however, do not provide full solutions. Have you ever heard a saying

"Give a man a fish, and he'll eat for a day. But teach a man how to fish, he'll eat for a lifetime"?

The same rule applies here, providing complete solutions can trick both of you, and your professor into thinking that you have enough skill to solve the problem by yourself, while in fact, you don't. And what if you are in the exam room? Will there be anyone there to actually help you?

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Ok, so your sequence is obviously bounded:
0 < q_n < 1.

Now, let's think about it, what if your sequence is monotonic? Then, it'll have limit as n tends to infinity, right?

So, let's try the following steps:

• First, find some initial values of the sequence that satisfies the requirements.
• Then, guess whether it's increasing or decreasing.
• Finally, try to prove it. If you don't know where to start, then Proof by Contradiction is the way to go. :)