Hilbert Sequence Space is Arc-Connected

Theorem
Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent.

Let $\ell^2 = \left({A, d_2}\right)$ be the Hilbert sequence space on $\R$.

Then $\ell^2$ is arc-connected.

Proof
Let $x = \left\langle{x_i}\right\rangle$ and $y = \left\langle{y_i}\right\rangle$.

Consider the mapping $f: \left[{0 \,.\,.\, 1}\right] \to \ell^2$ defined as:
 * $\forall t \in \left[{0 \,.\,.\, 1}\right]: f \left({t}\right) = t x + \left({1 - t}\right) y = \left\langle{t x_i + \left({1 - t}\right) y_i}\right\rangle$

which is convergent.

Then $f$ is an injective path joining $x$ to $y$.

Hence the result.