$\lim_{n\to\infty}\sum_{i=1}^n a_i + b_i = c$ and $\lim_{n\to\infty}\sum_{i=1}^n a_i = 0$ $\implies$ $\lim_{n\to\infty}\sum_{i=1}^n b_i = c$

Question

Let $a_i,b_i,c$ be real numbers. Suppose $\lim_{n\to\infty}\sum_{i=1}^n a_i + b_i = c$ and that $\lim_{n\to\infty}\sum_{i=1}^n a_i = 0$. Is it true that $\lim_{n\to\infty}\sum_{i=1}^n b_i = c$?

Answer 1

Yes, if $u_n$ and $v_n$ are two sequences which converge, $lim_n(u_n-v_n)=lim_nu_n-lim_nv_n$, write $u_n=\sum_{i=1}^{i=n}a_i+b_i$, $v_n=\sum_{i=1}^{i=n}a_i$.