I want to prove that if a Cauchy sequence of rational numbers is modified by changing a finite number of terms, then the result is an equivalent Cauchy sequence. I tried the problem but I'm not sure if what I'm saying is enough. Here it is:
Proof:
Let $\{y_n\}$ be the sequence obtained by modifying a finite number of terms in $\{x_n\}$. Since there are finitely many modified terms, they all must lie between the first term and the last modified term. Suppose a term $\{x_n\}$ is modified if $1 \leq n \leq N'$, where $x_{N'}$ is the last modified term of $\{x_n\}$.
Claim: $\{y_n\}\sim \{x_n\}$
Let $\epsilon \in \mathbb{Q}^+$ and choose $N \geq N' + 1$, which we may do by the archimedean property.
$|y_n-x_n| = |y_n - x_m + x_m - x_n| \leq |y_n - x_m| + |x_m-x_n|$, or
$|y_n-x_n| \leq |y_n - x_m| + |x_n-x_m|$.
Now, since $\{x_n\}$ is Cauchy and $y_n = x_n$ if $n \geq N' + 1$, it follows that:
$|y_n-x_n| \leq |x_n-x_m| + |x_n-x_m| \leq \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.
Therefore $\{y_n-x_n\}$ converges to zero and $\{y_n\}\sim \{x_n\}$.
Is this correct? Do I still need to show that $\{y_n\}$ is Cauchy?