Hilbert Cube is Arc-Connected

Theorem
Let $M = \left({I^\omega, d_2}\right)$ be the Hilbert cube.

Then $M$ is an arc-connected space.

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

Consider the mapping $f: \left[{0 \,.\,.\, 1}\right] \to I^\omega$ 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.