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Thread: Prove 2 statements on limits of sequences are true

  1. #1
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    Prove 2 statements on limits of sequences are true

    Hi, I have been told that the following two statements are true, but I don't have a clue how to actually prove it. I have been told that I can either prove it with words or with an example... :S

    1.- If the sequence ($\displaystyle a_n$) has a limit and the sequence ($\displaystyle b_n$) is such that $\displaystyle a_j=b_j$ whenever $\displaystyle j>10^{100}$ then the sequence ($\displaystyle b_j$) has a limit.
    2.- Every real number r is the limit of a sequence all of whose terms are rational numbers.
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  2. #2
    Super Member TheChaz's Avatar
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    1. The limit of $\displaystyle (a_n) = L$ if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have $\displaystyle | a_n - L | < \varepsilon$
    Well, the same goes for $\displaystyle (b_n)$! The "N" that we choose is just 10^100. So for n > 10^100, $\displaystyle (b_n) = (a_n)$, and therefore

    $\displaystyle |b_n - L| = |a_n - L|$, which we already know is less than epsilon.
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  3. #3
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    Quote Originally Posted by juanma101285 View Post
    1.- If the sequence ($\displaystyle a_n$) has a limit and the sequence ($\displaystyle b_n$) is such that $\displaystyle a_j=b_j$ whenever $\displaystyle j>10^{100}$ then the sequence ($\displaystyle b_j$) has a limit.
    2.- Every real number r is the limit of a sequence all of whose terms are rational numbers.
    For #1. To say that $\displaystyle (a_n)\to L$ means that almost all (all but a finite collection) of the $\displaystyle a_n's$ are 'close' to $\displaystyle L$. Does that mean almost all of the $\displaystyle b_n's $ must be close to $\displaystyle L~?$. HOW & WHY?

    For #2. Between any two numbers there is a rational number.
    Suppose $\displaystyle \alpha$ is a real number.
    For each positive integer $\displaystyle n$ there is a rational number $\displaystyle r_n$ between $\displaystyle \alpha~\&~\alpha+\frac{1}{n}$.
    Is it true that $\displaystyle (r_n)\to\alpha~?$
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  4. #4
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    Another way to look at (2): every real number can be written in decimal form: $\displaystyle x= A.a_1a_2a_3a_4...$

    Show x is the limit of the sequence A, $\displaystyle A.a_1$, $\displaystyle A.a_1a_2$, $\displaystyle A.a_1a_2a_3$, etc. Do you see that those are all rational numbers?

    For example, $\displaystyle \pi= 3.1415926...$ so it is the limit of the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ...
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