Prove or disprove if $x_k$ is unbounded and $|P|\geq 0$, then either $\lim_{k\to\infty}x_k^{\rm T}Px_k=0$ or $x_k^{\rm T}Px_k$ is unbounded

Question

Prove or disprove by counterexample:

For all $k\in\mathbb{N}$, let $x_k\in\mathbb{R}^{n}$, and assume $x_k$ is an unbounded sequence. Let $P$ be symmetric positive semidefinite. Then, either $\lim_{k\to\infty}x_k^{\rm T}Px_k=0$ or $x_k^{\rm T}Px_k$ is unbounded.

Answer 1

Suppose $P \neq 0$ and $\ker P$ is not trivial. Choose $x_0$ such that $x_0^TPx_0 = 1$ and a non zero $v \in \ker P$. Then $x_k=x_0+kv$ is unbounded but $x_k^T P x_k = 1$ for all $k$.