# Problem Concerning Cauchy Principle for Sequences.

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.

• What have you tried? – user10354138 May 21 at 14:16
• what do you mean by Cauchy Principle? – Mohammad Riazi-Kermani May 21 at 14:18
• @MohammadRiazi-Kermani by Cauchy Principle I mean the definition of a Cauchy Sequence. – Hayk Simonyan May 21 at 14:29
• @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. – Hayk Simonyan May 21 at 14:29
• $-a+2a=a$, $a+2a=3a$, $3a-a=2a\ge1$ – Mirko May 22 at 2:21