Definition of the problem
Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. Prove that for all $x,y\in E$ we have $ \left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\left\langle x,y\right\rangle \leq\left\Vert x+y\right\Vert \cdot\left\Vert x\right\Vert \left\Vert y\right\Vert . $
My efforts
I tried to apply Cauchy-Schwarz inequality: $ \left|\left\langle x,y\right\rangle \right|\leq\left\Vert x\right\Vert \left\Vert y\right\Vert \quad\forall x,y\in E, $ and since we are in an inner product space over $\mathbb{R}$, we can simplify remove the absolute value from the inner product: $ \left\langle x,y\right\rangle \leq\left\Vert x\right\Vert \left\Vert y\right\Vert \quad\forall x,y\in E. $ We obtain: $ \left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\left\langle x,y\right\rangle \leq\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\cdot\left\Vert x\right\Vert \left\Vert y\right\Vert \quad\forall x,y\in E. $
My question
Could you give me a hint/idea on how to solve this problem? Which Lemma/Theorem should I use?
Thank you,
Franck