# Prove $\lim_{n\rightarrow\infty}(a_n + b_n) = a + b$ if $\lim_{n\rightarrow\infty}a_n=a$ and $\lim_{n\rightarrow\infty}b_n=b$

I just need my solution checked, because I'm not actually sure if it proves what the question is asking, thanks! :)

Question:

Suppose $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are convergent sequences of real numbers with $\lim_{n\rightarrow\infty} a_n = a$ and $\lim_{n\rightarrow\infty} b_n = b$. Prove that:

$\lim_{n\rightarrow\infty}(a_n+b_n) = a+b$

Solution:

Fix $\epsilon>0$

Suppose $|a_n+b_n|<\epsilon \implies \exists\space \delta_1>0$ and $\delta_2>0$

$\therefore |a_n+b_n-a-b|<\epsilon$

$|a_n-a+b_n-b|<\epsilon$

Since $||x+y||\leq||x||+||y||$ (triangle inequality)

$\therefore |a_n-a|+|b_n-b|<\epsilon$

Let $|a_n-a|<\frac{\epsilon}{2}$

Let $|b_n-b|<\frac{\epsilon}{2}$

Let $\delta = \min{(\delta_1, \delta_2)}$

$\therefore |a_n-a|+|b_n-b|<\frac{\epsilon}{2}+\frac{\epsilon}{2} = \epsilon$

Note:

As I said above, I just need it checked to see if it actually proves what the question is asking, if not or there's an error could you please tell me how I could improve it, change it or fix it. Thanks heaps! :)

• Not sure what the $\delta$'s are for. Convergence of sequences has to do with what the sequences do at infinity, i.e. for very large $n$. So your proof should start with something like, $a_n \to a \Longrightarrow \exists N$ s.t. $n > N \Longrightarrow |a_n - a| < \frac{\epsilon}{2}$. But you are correct in trying to use the triangle inequality. Mar 30 '17 at 7:06

The ideas are kind of right, but your notation is all over the place, and the actual content is also wrong. (What even are your $\delta$s?)
Your second line: you've written "Suppose that if $|a_n + b_n| < \epsilon$, then there is $\delta_1 > 0$ and $\delta_2 > 0$." Your third line: "Therefore $|a_n + b_n - a-b| < \epsilon$."
This pair of lines isn't even grammatically correct, let alone mathematically correct. In your second line, you needed to let $N$ be such that for all $n > N$, we had $|a_n + b_n - a - b| < \epsilon$.
The fifth and sixth lines are not grammatically correct: "Since [thing], therefore [thing]" should be "Since [thing], we have [thing]" or similar. They're also not mathematically correct: if $|x-y| < \epsilon$, it's not necessarily true that $|x| + |y| < \epsilon$.
Then in your eighth and ninth lines, you implicitly redefine $n$ so now it has to be bigger than some new $N$.