Since $(\mathbb{N},\mathcal{T}_u\vert_\mathbb{N})=(\mathbb{N},\mathcal{T}_D)$ is complete and $\sum\limits_{n\in\mathbb{N}} n=-1/12$ is convergent in $(\mathbb{R},\mathcal{T}_u)$, the sequence of its partial sums is Cauchy in the subspace $(\mathbb{N},\mathcal{T}_u\vert_\mathbb{N})=(\mathbb{N},\mathcal{T}_D)$ and thus convergent. That is, $$\therefore-\frac{1}{12}\in\mathbb{N}$$