convergence of a sequence of inner products

Question

Let $a \in \mathbb{R}^n$ be non zero and let $\left\{ x_k, k \in \mathbb{N} \right\}$ be a sequence in $\mathbb{R}^n$. If $a^Tx_k$ converges to some real number, say $r$, does $x_k$ necessarily converge in $\mathbb{R}^n$?

Answer 1

Obviously no. Consider the following counter-example: $n=2$. $a=(0,1)$ which is a non-zero vector. $x_{n}=((-1)^{n},0)$. Clearly $\langle a,x_{n}\rangle=0$ for each $n$ but $(x_{n})$ does not converge to any $x\in\mathbb{R}^{2}$.

Answer 2

No. Let $a = (1,1)$ and $ x_k = \begin{cases} (1,-1) & \text{if } k \text{ is odd} \\ (-1,1) & \text{if } k \text{ is even} \end{cases}. $

Then $a \cdot x_k = 0$ for all $k$ but the sequence $x_k$ doesn't converge in $\mathbb{R}^2$.