# Proof of sum rule for null sequences

I have written the following proof and wanted to know if there are any problems with it or whether it is actually a valid proof. Please read through it and try and pick out any errors or points I may have overlooked. We are trying to prove that the sum of two null sequences (sum term wise) is null.

We have two sequences $a_n$ and $b_n$ where $(a_n)\rightarrow 0$ and $(b_n)\rightarrow 0$. By definition, $$\forall \epsilon >0, \exists N_1\in \mathbb{N} \text{ s.t } \forall n>N_1, |a_n|<\epsilon.$$ We can use the same definition for $b_n$ so that $|b_n|<\epsilon$ when $n>N_2.$ Consider $|ca_n+db_n|$ where $c$ and $d$ are real numbers. We have $$0\leq|ca_n+db_n|\leq |c||a_n|+|d||b_n|<(|c|+|d|)\epsilon ,$$ using the definitions above and the traingle ineqaulity.

As the value of $\epsilon$ is chosen arbitrarily it shows that $(ca_n+db_n)\rightarrow 0. \ \ \ \square$

Is this given proof ok or not?

• If ti is the same as your other question with $0$ instead of $a$ and $b$, then it is o.k. – user228113 Oct 19 '16 at 21:29
• It's slightly different above because I assumed this rule in that proof but here I am trying to fully justify the rule for null sequences only using definitions of a sequence tending to 0. – Anon Oct 19 '16 at 21:32
• When asking for proof verification, it helps to explicitly state what you are proving first. – arkeet Oct 19 '16 at 21:33
• Good point :)I shall add that in. – Anon Oct 19 '16 at 22:12