Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication. 

For $a\in V$ we define the "associated $\zeta$-function" to be \begin{eqnarray} \zeta(a) := \sum_{n=1}^\infty \frac{1}{n^{1+|a_n|}}
\end{eqnarray}
(similar to post https://mathoverflow.net/questions/187610/almost-zeta-function)

Let $D = \{a\in V: \zeta(a) = \infty\}$. Is $D\subseteq V$ a subspace?