Let's say we're in a Hilbert Space $H$, and have a sequence of vectors $||v_{n}||$ in $H$ such that $\lim_{n \to \infty} ||v_{n}|| = \infty$. Fix $y \in H$. Is it true that in this case $\lim_{n \to \infty} (v_{n} \cdot y) = \infty$ as well?
My approach at prove this was looking at this as $\lim_{n \to \infty} ||v_{n}|| (\frac{v_{n}}{||v_{n}||} \cdot y)$, and noting that the $||v_{n}||$ on the outside would diverge. But the problem here is that the vector $\frac{v_{n}}{||v_{n}||}$ inside is changing as $n$ increases, meaning that a priori we have no way of knowing that $(\frac{v_{n}}{||v_{n}||} \cdot y)$ doesn't become arbitrarily small as $n$ increases, and thus we don't have divergence.
If $(e_n)$ is orthonormal then $\|ne_n\|=n \to \infty$ but $\langle ne_n , e_1 \rangle \to 0$.