Is every converging sequence the sum of a constant sequence and a null sequence? Let $a_n$ be any sequence converging to $a$ when $n \to \infty$. 
Can you rewrite $a_n$ so that it is the sum of two other sequences? $$a_n=b_n + c_n,$$ with $b_n=b$ for every $n \in \mathbb{N}$ and $c_n\to 0$ as $n\to \infty$.
In other words: Is a converging sequence ($a_n$) actually a null sequence ($c_n$) "shifted" by a constant ($b$)?
Or is there any counterexample where one is not allowed to do so?

Let $(a_n)$ be a sequence converging towards $a$ and let define the following sequences: $$\forall n\in\mathbb{N},b_n:=a,c_n:=a_n-a.$$
One has: $$\forall n\in\mathbb{N},a_n=b_n+c_n$$
and $(c_n)$ is converging towards $0$.
