Hilbert Cube is Arc-Connected

Theorem
Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.

Then $M$ is an arc-connected space.

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

Consider the mapping $f: \closedint 0 1 \to I^\omega$ 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.