Suppose we have a nx1 vector $v_n = [\frac{1}{\sqrt{n}} \frac{1}{\sqrt{n}} ...... \frac{1}{\sqrt{n}}]^T$ in R^n, does is converge or diverge?
I initially thought it would converge to $\vec0$ since each of its entry $\frac{1}{\sqrt{n}} \rightarrow 0$, but then $\Vert v_n - \vec0 \Vert = \sqrt{(\frac{1}{\sqrt{n}})^2 \cdot n} = 1$, which shows that v_n doesn't converge to 0.
So I'm a little confused now. Any hints?