Question

Task If (a sub_n) and (b sub_n) are cauchy sequences, show directly that (a sub_n + b sub_n) is also a cauchy sequence

Answer 1

Question 1. If $(a_n)$ and $(b_n)$ are Cauchy sequences, show directly that $(a_n + b_n)$ is also a Cauchy sequence.

Solution. Since $(a_n)$ and $(b_n)$ are Cauchy sequences, for any $\varepsilon > 0$ there are $N_1 \in \mathbb{N}$, such that $|a_n - a_m| < \frac{\varepsilon}{2}$ for all $m, n > N_1$, and $N_2 \in \mathbb{N}$, such that $|b_n - b_m| < \frac{\varepsilon}{2}$ for all $m, n > N_2$. Set $N = \max\{N_1, N_2\}$. Then for all $m, n > N$ we have

$$\left| \left(a _ {n} + b _ {n}\right) - \left(a _ {m} + b _ {m}\right) \right| = \left| a _ {n} - a _ {m} + b _ {n} - b _ {m} \right| \leq \left| a _ {n} - a _ {m} \right| + \left| b _ {n} - b _ {m} \right| < \frac {\varepsilon}{2} + \frac {\varepsilon}{2} = \varepsilon .$$

Thus, $(a_{n} + b_{n})$ is a Cauchy sequence.