# Thread: Convergence of a sequence

1. ## Convergence of a sequence

Hey all,

Unsure on how to structure the proof to the following:

Suppose that the sequence (an) tends to the limit A, while the sequence (bn) tends to the limit B. Prove that the sequence (an + bn) tends to A + B.

Any hints would be greatly appreciated.

2. [QUOTE=scorpio1;113799]Hey all,

Unsure on how to structure the proof to the following:

Suppose that the sequence (an) tends to the limit A, while the sequence (bn) tends to the limit B. Prove that the sequence (an + bn) tends to A + B.

Pick a natural number $\displaystyle N_1$ so for all n > $\displaystyle N_1$

$\displaystyle |a_n-A|< \frac{\epsilon}{2}$

pick a natural number $\displaystyle N_2$ so for all n> $\displaystyle N_2$

$\displaystyle |b_n-A|< \frac{\epsilon}{2}$

let $\displaystyle N_3$ be the max of $\displaystyle N_1$ and $\displaystyle N_2$

the go for it. for all n > $\displaystyle N_3$

$\displaystyle |a_n+b_n -A-B|=|(a_n-A)+(b_n-B)|$

use the triangle inequality and you have got it made