A sequence of real numbers $($$x_n$$)$ converges to $x$. Consider the following claims:
(i) The sequence $(x_{n+1}/x_n)$ converges to $1$.
(ii) The sequence $(x_{n+1} + x_n)$ converges to $2x$.
Here's what I think: (i) Can be shown to be false by taking the example of the sequence $2^{-n}$, as $\frac{2^{-(n+1)}}{2^{-n}}$ is $\frac{1}{2}$.
Now, since $x_{n+1}$ does not necessarily converge to $x$, $(x_{n+1} + x_n)$ also does not necessarily converge to $2x$. I am not sure what the mistake with my logic is. According to my teacher, (ii) is true.
For $ii$.
Let $\epsilon>0$ given. there exists an integer $N$ such that $$n> N \;\;\implies \;\; |x_n-x|<\frac{\epsilon}{2}$$ but $$n>N \implies n+1>N$$ and
$$|x_{n+1}+x_n-2x|=$$ $$|(x_{n+1}-x)+(x_n-x)|\le$$ $$|x_{n+1}-x|+|x_n-x|<\epsilon.$$
Thus $$\lim_{n\to+\infty}(x_{n+1}+x_n)=2x$$
i