Metric inequality in Hilbert cube.

Question

Let $x,y \in H^{\infty}$
(The Hilbert cube is the collection of all real sequences $x=(x_n$) with $|x_n|\leq 1)$
and $k \in \mathbb{N}$

Let $M_k= \max \{|x_1-y_1|,…,|x_k-y_k|\}$

And let: $d(x,y)= \sum_{n=1}^{\infty} 2^{-n} |x_n-y_n|$

Show:

$$2^{-k} M_k \leq d(x,y) \leq M_k + 2^{-k+1}$$

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I have only succeeded in proving that $2^{-k}M_k\leq 1$ but further than that I am struggling. Any hints?

(Sorry to the ones that tried to solve $d(x,y) \leq M_k + 2^{-k}$, I was just informed that there is a typo in the book.)

Answer 1

$$ \sum_{n=1}^\infty 2^{-n} |x_n - y_n| = \sum_{n=1}^k 2^{-n} |x_n - y_n| +\sum_{n=k+1}^\infty 2^{-n} |x_n - y_n| .$$