What can I say about the consecutive difference of a convergent sequence?

Question

Suppose sequence $x_r\to 0$. Then what can I say about sequence $x_{r+1}-x_r$?does it converge to zero?

Answer 1

Hint: Convergent sequences are Cauchy, thus...

Answer 2

Given $\epsilon>0$

$$x_r\to x \implies \exists N\ge 0 \; : $$ $$\; \forall r\ge N \;\; |x_r-x|<\frac{\epsilon}{2}$$

but for $r\ge N$, we have

$$r+1>r\ge N$$ thus for $r\ge N,$

$$|x_{r+1}-x_r|\le |x_{r+1}-x|+|x_r-x|<\epsilon$$

done.

Answer 3

The fact that $x_{r}\rightarrow0$ is irrelevant, so let's proceed assuming $x_{r}\rightarrow x$ (i.e., the sequence converges).

Then, $$ \lim_{r\rightarrow\infty}\left\{ x_{r}-x_{r+1}\right\} =\lim_{r\rightarrow\infty}x_{r}-\lim_{r\rightarrow\infty}x_{r+1}=x-x=0. $$ In the above, we used the fact that $x_{r+1}\rightarrow x$ and that the limit of sums is the sum of limits.