I am reading the book Putnam and Beyond by R.Gelca and T.Andreescu, and there is the following problem (#118):
Let $V_1,V_2,...,V_m$ and $W_1,W_2,...,W_m$ be isometries of $\mathbb{R}^n$ ($m,n$ positive integers). Assume that for all $x$ with $||x|| \leq 1$, $||V_i x - W_i x|| \leq 1,\ i = 1,2,...,m$. Prove that: $$ \Big|\Big| \big(\prod_{i=1}^m V_i\big)\ x - \big(\prod_{i=1}^m W_i\big)\ x \Big|\Big| \leq m $$ for all $x$ with $||x|| \leq 1$.
Why the following solution is wrong (I presume that it's wrong because authors used a more sophisticated one and mine is too obvious):
We know, that isometries form a group, so $V = V_2 \cdot ... \cdot V_m$, $W = W_2 \cdot ... \cdot W_m$ are isometries too. That's why for all $x$ such that $||x|| \leq 1$ we have $||Vx|| = ||x|| \leq 1$ and $||Wx|| = ||x|| \leq 1$. But this means that vectors $Vx$ and $Wx$ satisfy the property from the task and we have $||V_1(Vx) - W_1(Wx)|| \leq 1$, which is the desired result (and even stronger).
P.S. If my argument is valid, then what condition should be added to the task so it is not valid anymore?
You can only conclude $$\lVert V_1(Vx) - W_1(Wx) \rVert = \lVert V_1(Vx) - V_1(Wx) + V_1(Wx) - W_1(Wx) \rVert \le \lVert V_1(Vx) - V_1(Wx) \rVert + \lVert V_1(Wx) - W_1(Wx) \rVert = \lVert Vx - Wx \rVert + \lVert V_1(Wx) - W_1(Wx) \rVert \le\lVert Vx - Wx \rVert + 1.$$
This reduces a product of $m$ isometries to one of $m-1$.