Induction proof summing sequences am trying to extend the following to k terms:  $a_{n}\rightarrow 0$ and $b_{n} \rightarrow 0, \alpha,\beta \in \mathbb{R}$ then $(\alpha a_{n} + \beta b_{n}) \rightarrow 0$.
Attempt:
We use induction. For the base case we have that $|a_{n}| < \frac{\epsilon}{2|\alpha|}, |b_{n}| < \frac{\epsilon}{2|\beta|}$ so that $(\alpha a_{n} + \beta b_{n}) \rightarrow 0$.
Inductive step: Assume that we sum $k$ terms i.e. K = $(\alpha a_{n} + \beta b_{n} + \dots \kappa k_{n}) \rightarrow 0$.
Final step consider $K+1 = K + \lambda l_{n}$. Then we have we know that $|a_{n}| < \frac{\epsilon}{N+1|\alpha|}, |b_{n}| < \frac{\epsilon}{N+1|\beta|}, \dots (i.e. N+1)$ terms here
so if we sum all of the terms together plus the $\lambda l_{n}$ term we have$|\alpha a_{n}+\beta b_{n} + \dots \lambda l_{n}| \leq |\alpha||a_{n}|+|\beta||b_{n}| + \dots |\kappa||k_{n}| + |\lambda||l_{n}| \leq \epsilon$
I know the ending argument isn't written the cleanest. I know what the argument is roughly (as shown) but struggling to right the final step cleanly / formally.
Any help appreciated.
 A: If induction is a requirement, then here is a proof:
Let $S$ be the set of all natural numbers $k$ such that if there are $k$ sequences $a_{1,n},a_{2,n},\dots,a_{k,n}$ such that $\lim\limits_{n\to\infty}a_{j,n}=0$ for all $1\leq j\leq k$ then for scalars $c_1,c_2,\dots,c_k$, we have $$\lim\limits_{n\to\infty}\sum_{i=1}^{k}c_ia_{i,n}=0$$
We want to show that $S$ is inductive.
Base Case:
I leave this to you, as all you need to show is if $a_{1,n}$ is a sequence that converges to $0$, then $\lim\limits_{n\to\infty}c_1a_{1,n}=0$.
Inductive Hypothesis:
Suppose that if there are $k$ sequences $a_{1,n},a_{2,n},\dots,a_{k,n}$, all converging to $0$, then for scalars $c_1,c_2,\dots,c_k$, we have $$\lim\limits_{n\to\infty}\sum_{i=1}^{k}c_ia_{i,n}=0$$
Inductive Step:
Suppose that we have $k+1$ sequences $a_{1,n},a_{2,n},\dots,a_{k,n},a_{k+1,n}$ all converging to $0$. Suppose we have scalars $c_1,c_2,\dots,c_{k},c_{k+1}$. We have $\lim\limits_{n\to\infty}c_{k+1}a_{k+1,n}=0$. By hypothesis, we have
$$\lim\limits_{n\to\infty}\sum_{i=1}^{k}c_ia_{i,n}=0$$ Since for any convergent sequences $\alpha_n,\beta_n$ converging to $\alpha$ and $\beta$ respectively, we have $\lim\limits_{n\to\infty}(\alpha_n+\beta_n)=\alpha+\beta$, then we are done, as
$$\begin{align*}
\lim\limits_{n\to\infty}\sum_{i=1}^{k+1}c_ia_{i,n}
&=\lim\limits_{n\to\infty}\left(\sum_{i=1}^{k}c_ia_{i,n}+c_{k+1}a_{k+1,n}\right)\\
&=\lim\limits_{n\to\infty}\sum_{i=1}^{k}c_ia_{i,n}+\lim\limits_{n\to\infty}c_{k+1}a_{k+1,n}\\
&=0+0\\&=0
\end{align*}$$
