Cauchy sequences are convergent, part of the proof verification

I just want to confirm that one particular reasoning in the proof of the theorem is correct

Let $x_n$ be a Cauchy sequence, and let $k,N \in \mathbb N, m=N$ and $k \geq N$. Then since $x_n$ is Cauchy we have $$|x_k - x_N|<\frac{\varepsilon}{3} \implies x_N - \frac{\varepsilon}{3} < x_k < x_N + \frac{\varepsilon}{3} \quad (1)$$which guarantees us that the Cauchy sequence is bounded, with the following argument from Zorich's Mathematical Analysis

but since only a finite number of terms of the sequence have indices not largerthan $N$, we have shown that a Cauchy/fundamental sequence is bounded.

Now, for $n \in \mathbb N$ define $a_n = \inf\limits_{k\geq n}x_k$ and $b_n = \sup\limits_{k\geq n}x_k$.

Also note that $$a_n \leq b_n \quad \forall n \in \mathbb N \quad (2)$$

The following reasoning and the bolded statements are what I want to verify:

Then from $(1), (2)$ and because $a_n$ and $b_n$ are subsequences of $x_n$ we have $$x_N - \frac{\varepsilon}{3} < a_n \leq b_n < x_N + \frac{\varepsilon}{3} \quad \forall n > N$$

and in particular because $a_n$ and $b_n$ for $n > N$ are subsequences of the subsequence $x_k$ for $k >N$.