Hilbert cube

The Hilbert cube is homeomorphic to countably many copies of the unit interval $[0,1]$, sometimes written as $H=[0,1]^{\infty}$.
Proposition: The Hilbert cube is metrizable; define a metric for $\vec{x}=(x_1,x_2,\ldots)$ and $\vec{y}=(y_1,y_2,\ldots)$ in the Hilbert cube by $$d_{\infty}(\vec{x},\vec{y})=\displaystyle\sum_{k=1}^{\infty} \dfrac{|x_i-y_1|}{2^k}.$$
Proof: We must prove that $d$ is a metric. █