sequences and series - Let ${a_n},{b_n}>0 ,lim limits_{n to infty} [a_n+b_n]=0 $ then$ lim limits_{n to infty}a_n=0 $ and$ lim limits_{n to infty} b_n = 0$

Let ${a_n}$ and ${b_n}$ be sequences of nonnegative numbers. Show that if

$\lim \limits_{n \to \infty} [a_n+b_n]=0$ then $\lim \limits_{n \to \infty}a_n=0$ and $\lim \limits_{n \to \infty} b_n = 0$.

I feel like this is just obvious because the only way you can add nonnegative numbers and have them equal $0$ is if both the numbers are $0$. But I don't really know what I'm supposed to show...

Answer

We have that $\forall \epsilon>0. \; \exists N \in \mathbb N. \; \forall n > N. \; |a_n+b_n-0|=|a_n+b_n|<\epsilon$.

Because $a_n \ge0$, $b_n\ge0$, $|a_n+b_n|\ge|a_n|$, $|a_n+b_n|\ge|b_n|$.

Therefore, $|a_n-0|<\epsilon$, $|b_n-0|<\epsilon$ when $n>N$.

Hence $\lim_{n\rightarrow\infty}a_n=\lim_{n\rightarrow\infty}b_n=0$.