Let $h_1, h_2, \cdots$ be an orthonormal basis for Hilbert space $H$. Let $h$ be any vector in $H$. Prove that there exists a sequence $c_1, c_2, \cdots$ of complex numbers such that the sequence $\underline{h}_1, \underline{h}_2, \cdots$ in $H$, with $\underline{h}_n=c_1 h_1+c_2 h_2 +\cdots+c_n h_n$, approaches the vector $h$. Prove that the choice $c_n=\left(h_n, h\right)$ is the unique one for this to be true.