# $\lim_{n\rightarrow \infty} (a_n-b_n)=0$ and $\lim_{n\rightarrow \infty} b_n=0$ imply $\lim_{n\rightarrow \infty}a_n=0$?

Consider two sequences $\{a_n\}_{n\in \mathbb{N}}\in [0,1]$ and $\{b_n\}_{n\in \mathbb{N}}>0$ and suppose $$\begin{cases} \lim_{n\rightarrow \infty} (a_n-b_n)=0\\ \lim_{n\rightarrow \infty} b_n=0\\ \end{cases}$$ Does this imply $\lim_{n\rightarrow \infty}a_n=0$?

Yes, because$$\lim_{n\to\infty}a_n=\lim_{n\to\infty}\bigl((a_n-b_n)+b_n\bigr)=\lim_{n\to\infty}(a_n-b_n)+\lim_{n\to\infty}b_n=0+0=0.$$
An alternative: assume towards a contradiction that $\lim_{n\rightarrow\infty} a_n = c$ for some $c\neq 0$
Then we'd have $\lim_{n\rightarrow\infty} (a_n - b_n) = \lim_{n\rightarrow\infty} a_n - \lim_{n\rightarrow\infty}b_n = c - 0 = c\neq 0$