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Given $\{{a_n}\}_{n=0}^{\infty}$ and $\{{b_n}\}_{n=0}^{\infty}$

Prove or disprove:

1) if $\lim \limits_{n\to \infty}a_n=0$ then $\lim \limits_{n\to \infty}a_n-[a_n]=0$

I think (1) is correct because if $\lim \limits_{n\to \infty}a_n=0$ then by the definition of limit I can show that for each $\epsilon>0, |a_n-[a_n]|<\epsilon$

2)If $a_n$ converges and $b_n$ doesn't converge then $(a_n+b_n)$ doesn't converge.

I think (2) is correct, but I'm not sure how to start proving it - maybe I can assume that it isn't correct and then get a contradiction?

3)If $\lim \limits_{n\to \infty}\frac{a_n+a_{n+1}+a_{n+2}}{3}=0$ then $\lim \limits_{n\to \infty}a_n=0$

I have no idea about (3).

My knowledge is of simple calculus theorem(limit definition, arithmetics of limits and the Squeeze Theorem).

Thanks a lot for your time and help.

  • 2
    Negative numbers can converge to zero, right?2012-03-30
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    @ GEdgar right. the sequence -1/n converges to zero.2012-03-30
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    And what about $a_n - [a_n]$ for that sequence?2012-03-30
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    The author might mean rounding to integers instead of floor/ceil...2012-03-30
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    i think before you try to proof something you should try to find some counterexamples. Sums can vanish despite the fact that the absolute value of ther summands is large (2 and 3). If $\lim a_n=0$ then ${abs}(a_n)<0.5$ for $n>N$. Then there are not much possibilities for $\[a_n\]$2012-03-30
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    Please try not to have multiple disparate questions in one. They might all be part of one homework, but they are essentially different. Of course, you do have to consider that posting multiple questions is bad too.2012-03-30
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    @Aryabhata actually I'm glad you said that - I thought that it will not be OK to ask it in multiple threads. Thanks for comment.2012-03-30
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    $a_n-[a_n]$ converges to -1.2012-03-30

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