Let $x_n$ be a sequence such that $|x_n −x_{n+1}|\leq 1/n$. Does this sequence always converge?
Show that $x_n = \sqrt{2 + x_{n−1}}$ is bounded and increasing.
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Do you know what is a Cauchy sequence? – Oct 13 '15 at 11:50
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3For the first problem consider the harmonic series. – DRF Oct 13 '15 at 11:51
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Your second sequence is not increasing if you start with an initial value greater than 2, in opposite it will be strictly decreasing. It is increasing for initial values between -2 and 2. – AmirHosein Sadeghimanesh Oct 13 '15 at 13:23
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Please, post only one question in one post. Posting several questions in the same post is discouraged and such questions may be put on hold, see meta. – Martin Sleziak Oct 13 '15 at 13:42
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You can find many posts about the second problem. Just have a look at this post and other questions linked there. – Martin Sleziak Oct 13 '15 at 13:43
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This might help you with the first part: Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?. – Martin Sleziak Oct 14 '15 at 13:28
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Hint: For the first problem consider the harmonic series.
Before you start the second part check whether you possibly have some extra information about $x_1$. I'm pretty sure you must otherwise you might want to consider say $x_1=100$.
For the second part consider first the increasing part. You should find that quite easy just by writing the appropriate inequality.
To show it's bounded consider at what starting point you stop having the increasing property.
DRF
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