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Math Help - 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 ( a_n) has a limit and the sequence ( b_n) is such that a_j=b_j whenever j>10^{100} then the sequence ( 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 (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  | a_n - L | < \varepsilon
    Well, the same goes for (b_n)! The "N" that we choose is just 10^100. So for n > 10^100, (b_n) = (a_n), and therefore

    |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 ( a_n) has a limit and the sequence ( b_n) is such that a_j=b_j whenever j>10^{100} then the sequence ( 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 (a_n)\to L means that almost all (all but a finite collection) of the a_n's are 'close' to L. Does that mean almost all of the b_n's must be close to L~?. HOW & WHY?

    For #2. Between any two numbers there is a rational number.
    Suppose \alpha is a real number.
    For each positive integer n there is a rational number r_n between \alpha~\&~\alpha+\frac{1}{n}.
    Is it true that (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: x= A.a_1a_2a_3a_4...

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

    For example, \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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