I am having trouble understanding the proof of theorem $19.2$ in "Elementary Analysis-The Theory Of Calculus" By Kenneth A. Ross, here it is:
I don't understand the purpose of replacing $\delta$ with $\frac{1}{n}$, I know that it is mathematically correct since $\frac{1}{n}>0$ when $n>0$ but I can't see how this added any value to the proof.
The second question, does $\lim_{k\Rightarrow \infty} y_{n_k}=x_0$ stem from the fact that $|x_n-y_n|<\delta$ which implies they are convergent to the same limit, hence all their subsequences converge to this same limit as well?
Best Answer
This is needed to move from statements about any positive real $\delta$ to get sequences. We take the sentence "For each $\delta > 0$ ..." and plug in $\delta = 1, \frac{1}{2}, \frac{1}{3}$ to show that the sequences $\{x_n\}$ and $\{y_n\}$ with property $|x_n - y_n| < \frac{1}{n}$ exist. Leaving it as $\delta$, we would have no relation to the sequence index, and having differences smaller than any constant $\delta$ doesn't make the difference converge.
Almost. But again, $|x_n - y_n| < \delta$ doesn't describe convergence; we need the difference to get smaller as a function of $n$, which is true in $|x_n-y_n| < \frac{1}{n}$.
To be more complete about it, the convergence of $\{x_{n_k}\}$ means that for any $\epsilon > 0$ we can find $K \in \mathbb{N}$ such that $k>K$ implies $|x_{n_k} - x_0| < \frac{\epsilon}{2}$. So whenever $k$ is large enough that $k > K$ and $n_k > \frac{2}{\epsilon}$ then by the triangle inequality,
$$ |y_{n_k} - x_0| \leq |y_{n_k} - x_{n_k}| + |x_{n_k} - x_0| < \frac{1}{2/\epsilon} + \frac{\epsilon}{2} = \epsilon $$
which shows $\lim_{k \to \infty} y_{n_k} = x_0$.