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I came across the following problems on convergence of sequences during the course of my self-study of real analysis:

Suppose $a_n \to a$. Define $$s_n = \frac{1}{n}\sum_{k=1}^{n} a_k$$ Prove that $s_n \to a$.

So $(a_n-a)$ is a null sequence. I want to show that $(s_n-a)$ is a null sequence. By a previous exercise, I know that $(x_n)$ is a null sequence $\implies$ $(y_n)$ is a null sequence where $y_{n} = (x_1+ \cdots+ x_n)/n$. So can we do something analogous to "adding $a$ to both sides" to get the desired result?

Show that the sequence $$a_n = \left(1- \frac{1}{2} \right) \left(1- \frac{1}{3} \right) \cdots \left(1- \frac{1}{n+1} \right)$$ is convergent.

So $a_1 = \frac{1}{2}$, $a_2 = \frac{1}{3}, \dots, a_n = \frac{1}{n+1}$. So I conjecture that $(a_n)$ is a null sequence. In other words, for each $\epsilon >0$, $|a_n| \leq \epsilon$ for all $n>N$. Let $\epsilon = \frac{1}{n}$. Choose $N = n+1$. Then the convergence follows?

Prove that the sequence $$a_n = \frac{1}{n+1}+ \frac{1}{n+2} + \dots + \frac{1}{n+n}$$ is convergent to a limit $\leq 1$.

So $a_{n} < a_{n+1}$ for all $n$. Then I need to show that it is bounded above by $1$. To show this should I consider $(1-a_n)$? All the terms are $<1$.

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    Damien, do you realize you've posted 17 questions in the span over the last two days, each beginning with "I came across this problem...)? To others, note that in two day's time Damien has earned 365 points by asking 17 questions. Others who post more elementary questions this frequently rarely receive this kind of positive reinforcement!2011-06-23
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    @amWhy: Yes. Should I write something else?2011-06-23
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    @amWhy: One reason is that Damien actually seems to have shown efforts to solve each problem he has asked so far, the questions are well formatted and he shows genuine interest in learning. Of course, by taking it slower (i.e. giving more thought rather than asking here) would only help Damien and he should consider asking only when he is really really stuck. I agree with that, if that is where you are coming from.2011-06-23
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    @amWhy: I have no teacher to check my work. Also the book doesnt have solutions. Hence I post some problems here.2011-06-23
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    Damien: There's nothing wrong about asking questions; that's what this site is for. But I become a little concerned when one user posts as many questions as you have in such short a period of time. Whether self-study or homework...I don't know if its such a good idea to become too "dependent" on others for solutions. To your credit, you do show work, speculate, etc., and we really appreciate that. Sometimes struggling a little while longer with a problem before asking for help can go a long way...and helps you better clarify the concepts that are *really* problematic for you?2011-06-23
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    Yes, @Aryabhata...see my follow up post which I was writing before I saw your comment. I did take note: to Damien's credit, he really has done a good job of showing work, speculating, engaging...etc. To Damien: I understand; it can get really frustrating when you have no where to turn. I'm not suggesting you stop posting questions; not at all. But sometimes working through a set of exercises, first, helps you identify conceptual difficulties, and choosing one or two exemplary exercises to ask about may prevent you from asking about "A?", then having to post a follow up "A II?"2011-06-23
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    Damien: what text are you using? Perhaps we can recommend a supplementary text which has more examples, and some solutions to its exercises which might help you work through your text; also, if you try googling using the text title, author's name, and, say "lecture notes", you will likely encounter course material, notes, practice tests, solutions, available at a real analysis course website for which the required text is the text you are working from.2011-06-23
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    @Damien: regarding your question about whether you should write something else as a "lead in" to your question. The tags: , and should be sufficient to inform us of those aspects; so you can pretty much start with your question, or perhaps add a preliminary comment about the section of the text your dealing with; that sort of detail helps us better tailor answers to where you're at, rather than telling you what you already know, or perhaps worse, providing a solution that involves material/theorems you haven't covered yet.2011-06-23
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    @amWhy: But I don't want to write in the imperative tone. Thus I start my questions like that.2011-06-23
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    @amWhy: I don't think self-learning is an appropriate tag. It is more appropriate for questions like "which books do I need?" etc. Almost every question can be tagged self-learning, if you use it that way!2011-06-23
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    @Aryabhata: sorry then, about the suggestion to use the self-learning tag. @Damien: I guess the "self learning" tag isn't a great idea. A brief mention that "I'm studying t analysis on my own ... and wondering about this problem/...and I'm stuck on this problem... and I don't understand this problem, (proof, exercise...)...I'm not sure where to start with... Not a big deal, really...And I wouldn't worry too much, because your questions have been consistently well-stated, nicely formatted, etc.2011-06-23
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    @amWhy: No worries. In general, tagging with "meta" tags is usually discouraged, as they normally don't give an idea as to what subjects the question could fall into. There are discussions about this on meta (both math.meta and meta.stackoverflow).2011-06-23

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