I have a question but can't seem to figure out how to solve it. The problem states:

Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a) if $x_n \geq 1$, then $(-1)^nx_n + 2x_n$ is divergent. (b) if $a = 0$, then $(-1)^nx_n + \frac1n$ is convergent.

I'd appreciate any help I can get. Thank you in advance.

  • 1
    $\begingroup$ What have you tried? $\endgroup$ – user10354138 May 21 at 14:16
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    $\begingroup$ what do you mean by Cauchy Principle? $\endgroup$ – Mohammad Riazi-Kermani May 21 at 14:18
  • $\begingroup$ @MohammadRiazi-Kermani by Cauchy Principle I mean the definition of a Cauchy Sequence. $\endgroup$ – Hayk Simonyan May 21 at 14:29
  • $\begingroup$ @user10354138 I tried to do something along the lines of the definition of a cauchy sequence. i.e. for all epsilon greater than 0 there exists a natural number n_0 and there exists 2 numbers n and m that are greater than n_0 such that the absolute value of x_n - x_m is less than epsilon. but I get stuck right there. Don't know what to do. $\endgroup$ – Hayk Simonyan May 21 at 14:29
  • $\begingroup$ $-a+2a=a$, $a+2a=3a$, $3a-a=2a\ge1$ $\endgroup$ – Mirko May 22 at 2:21

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