Let $H$ be a Hilbert space with norm $\|\cdot\|_X$ and scalar product $(\cdot,\cdot)_X$ , which is the direct sum of the subspaces $V$ and $W$. Regarding $V$ and $W$ it holds the intensified Cauchy Schwarz Inequality.
What do I mean with intensified Cauchy Schwarz Inequality? : There is a $\gamma$ with $0 \leq \gamma < 1$ and $|(v,w)|_X \leq \gamma \|v\|_X \|w\|_X \quad \forall v \in V, w \in W$
$P_V$ and $P_W$ are the orthogonal projections on $V$ and $W$. For a $u_0 \in X$ we compute $u_{2k-1} = P_V u_{2k-2}$ and $u_{2k} = P_W u_{2k-1}$.
Show: $\|u_n\|_X \leq \gamma^{n-1} \|u_0\|_X \quad \forall n \geq 1$
My thoughts: It is clear that we have use the Cauchy Schwarz inequality. I mean there is our $\gamma$. It seems that we have use the inequality multiple times to get $\gamma^{n-1}$.We also have to use the successive computation somehow. Maybe my idea is too simple but can induction help us?
Hint: We have $\lVert P_Wv\rVert^2=(v,P_Wv)_X$ and $\lVert P_Vw\rVert^2=(P_Vw,w)_X$.