Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of distinct vectors in $\mathbb{R}^4$ such that $\| x_n \|$ is convergent to $\alpha \in \mathbb{R}$. Is the sequence $(x_n)_{n\in \mathbb{N}}$ convergent?
I've found in a book that I could use the "diagonalization process". I didn't what this meant, so I searched "diagonalization process" and I found that it was a method used by Cantor when he proved that $(0,1)$ is uncountable. The thing is that I don't know how to apply this method in this specific problem. Can you explain this to me, please?
Consider the unit sphere with $\Vert x\Vert =1$. Clearly any sequence of points on this sphere has convergent norms, since the norms are all the same. Can you think of a non-convergent sequence of points on this sphere? (I can think of many by repeatedly going around the sphere).