A Real Hilbert Space Containing an Orthonormal Sequence Is Not Finite-Dimensional

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space, with its inner product $\langle\cdot,\cdot\rangle$, its norm $|\cdot|$ and its zero vector $0_{H}$ as fixed there, and let $\mathbb{N}$ be the set of natural numbers. Suppose that there is an orthonormal sequence $(f_{j})_{j\in\mathbb{N}}$ in $H$.

(i) Then $H$, which is a vector space over $\mathbb{R}$ by Real Inner Product Spaces, is not finite-dimensional.