Hilbert Sequence Space is Arc-Connected

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

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

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

Proof
Let $x = \sequence {x_i}$ and $y = \sequence {y_i}$.

Consider the mapping $f: \closedint 0 1 \to \ell^2$ defined as:
 * $\forall t \in \closedint 0 1: \map f t = t x + \paren {1 - t} y = \sequence {t x_i + \paren {1 - t} y_i}$

which is convergent.

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

Hence the result.