Show that $\lim_{n \to \infty} (x_n +y_n) = L + M$

Question

I am just wondering if I can take $ {\varepsilon}' = \varepsilon _a + \varepsilon _b $ where $|x_n - L| < \varepsilon _a $ and $|y_n - M| < \varepsilon_b$

If not could someone point me in the right direction. Thanks.

Answer 1

The standard way to deal with this is to say:

Given $\epsilon > 0$, there is an $N_1 > 0$ such that

$$n > N_1 \ \Longrightarrow \ |x_n - L| < \epsilon/2$$

and an $N_2 > 0$ such that

$$n > N_2 \ \Longrightarrow \ |y_n - M| < \epsilon/2$$

Can you finish from here?