Let $N_a$ be an arbitrary norm on $\mathbb{F}^n$ where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. I have a sequence of $x_k \in \mathbb{F}^n$ such that $|x_{1,k}|$ is unbounded above (in $\mathbb{R}$) (where $x_{1,k}$ is the first entry of $x_k$), how can I prove that $N_a(x_k)$ cannot be a sequence that is bounded above in $\mathbb{R}$?
I feel like I am missing something obvious. I am not allowed to use the fact that all norms on $\mathbb{F}^n$ are equivalent. Please give a hint.