Can we say $ \left\| \sum_{n=1}^{\infty} x_n\right\|_{X} \geq C \|x_N\|_{X}$?

Question

Let$ (X, \|\cdot\|_{X})$ be a Banach space, and $\{x_n\}\subset X$.

Can we say that $ \left\| \sum_{n=1}^{\infty} x_n\right\|_{X} \geq C \|x_N\|_{X}$ for some constant $C>0$ and some $N\in \mathbb N$?

Answer 1

Consider $(x_h)_h$ as $x_{2n}=\frac{(-1)^n}{n}$, and $x_{2n+1}=-x_{2n}$ . Then we end up with $0\geq C$, contradiction.

Thanks to the comments, I was totally wong.