In Introduction to Infinite Series by Bonar and Khoury, the following are given as "facts" but left up to the reader to prove. Can you please let me know if I made any errors in my attempts and if the proof is completely wrong give the correct proof? Also you are more than welcomed to give a shorter proof.
Theorem:
Changing a finite number of terms in a sequence has no effect on the convergence, divergence or the limit if it exists.
For example, the sequences
$$1, \frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7}\cdots , \frac{1}{n}, \cdots$$
and
$$ 2,7,5,\frac{1}{10},\frac{1}{5},\frac{1}{6}, \frac{1}{7}, \cdots , \frac{1}{n}, \cdots$$
both converge $0$.
Proof. By definition, a sequence converges to a real number $A$ if , for each $\epsilon>0$, there exists an integer $N$ such that for all $n>N$, $|a_n-A|<\epsilon$. As we can see from the definition changing any terms $|a_N|$ for all $N<n$ does not affect the limit and therefore convergence because $|a_N|$ is independent on the choice of $\epsilon$.
And by definition, a sequence diverges to $\infty$ if, for any $M>0$, there exists an integer $N$ such that all $n>N$ , $a_n>M$. Similarly, we can change any $a_N<M$, and it does not affect the choice of $\epsilon$ for $a_n$.A similar argument can be used for the case of divergence to $-\infty.$ $\blacksquare$
Best Answer
Your idea is right. You can make it a bit more formal as follows.
Suppose we start with a sequence $a_n$. If we change finitely many terms, then this results in a new sequence $b_n$. Since we only changed finitely many terms, there is some $M$ such that $a_n = b_n$ for all $n > M$.
Now suppose that $a_n$ converges to $A$. Let $\epsilon > 0$. There is some $N$ such that $|a_n - A| < \epsilon$ for all $n > N$.
Then, for all $n > \max\{N, M\}$, we have $b_n = a_n$, so $$|b_n - A| = |a_n - A| < \epsilon$$ which shows that $b_n$ also converges to $A$.
We have shown that if two sequences differ in only finitely many terms, and one sequence converges, then the other also converges (to the same limit). The contrapositive: if one diverges, then the other must also diverge.