[Math] If $\{a_n\}$ and $\{b_n\}$ are Cauchy, then $\{a_n + b_n\}$ is Cauchy.

Question

If $\{a_n\}$ and $\{b_n\}$ are Cauchy, then $\{a_n + b_n\}$ is Cauchy.

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Proof:

$|a_{m_1}-a_{n_1}|\lt \epsilon_1$ and $|b_{m_2} – b_{n_2}|\lt \epsilon_2$

Then take $m_3=\max(m_1,m_2),n_3=\max(n_1,n_2)$

Then $|a_{m_3}+b_{m_3} – a_{n_3}-b_{n_3}|\leq |a_{m_3}-a_{n_3}|+|b_{m_3}-b_{n_3}|\lt2\epsilon$

Now I am unsure how to progress. It would work if my original cauchy sequences were less than $\frac{\epsilon}{2}$, but I don't understand how I would obtain this. Thanks

Answer 1

Fix $\epsilon > 0$.

$a_n$ and $b_n$ are Cauchy, so $\exists N_1, N_2$ such that $|a_m - a_n| < \epsilon/2$ when $m, n > N_1$ and $|b_m - b_n| < \epsilon/2$ when $m, n > N_2$.

Choose $N = \max(N_1, N_2)$, then we have

$$|(a_m + b_m) - (a_n + b_n)| \leq |a_m - a_n| + |b_m - b_n| < \epsilon/2 + \epsilon/2 = \epsilon.$$

whenever $m, n > N$, so $\{a_n + b_n\}$ is Cauchy.