Theorem about orthogonal system in inner product space. It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$
satisfying 
$$
\langle x_n, x_m\rangle =0 \quad\forall n\ne m,
$$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if and only if $\displaystyle\sum_{n=1}^{\infty}\|x_n\|^2$ is convergent". I would like to know if $H$ is an inner product space (pre-Hilbert space) the above statement is still true?

No. Take the subspace $l^2_{\text{fin}}(\mathbb R)$ of $l^2(\mathbb R)$ of all real-valued sequences with only finitely many nonzero entries. $l^2_{\text{fin}}(\mathbb R)$ is a pre-Hilbert space with the induced inner product and the sequences $\delta_{i,j}$ whose only nonzero entry is $\delta_{i,i} = 1$ form an orthonormal system $\{ \delta_{i,j}\} \in l^2_{\text{fin}}(\mathbb R)$ but obviously $\sum_i 2^{-i} \delta_{i,j} \not \in l^2_{\text{fin}}(\mathbb R)$.
