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?

For a constant $b$ any number $a$ (whether it's a term in a sequence or not) can be written as $a = b + c$ where $c = a - b$ so any sequence $\{a_n\}$ can be written as $\{b + c_n\}$ where $c_n = a_n - b$ and in particular if $\lim a_n = a$ then the  sequence can be written as $\{a + c_n\}$ where $c_n = a_n - a$.  And clearly $\lim \{a_n\} = \lim \{a + c_n\} = a$.
So your question boils down to does $\lim\{b + c_n\} = b + \lim\{c_n\}$? And therefore if $b = a = \lim a_n$ does $\lim c_n = 0$.
This should be a basic proposition early on in the study of convergent sequence and the answer is: yes.
$|a - a_n| = |(a -b) - (a_n - b)| = |(a - b) - c_n|$.  So whatever $\epsilon$, $N$, $n > N$ crap that you can say about $a$ and $a_n$ can also be said about $(a-b)$ and $c_n$.  
So if $c_n = a_n - b$ then $\{a_n\} \rightarrow a \iff \{c_n\} \rightarrow (a - b)$.
