TITLE: Inner product space over $\mathbb{R}$
QUESTION [9 upvotes]: Definition of the problem
I have to prove the following statement:
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 
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\left|\left\langle x,y\right\rangle \right|\leq\left\Vert x+y\right\Vert \cdot\left\Vert x\right\Vert \left\Vert y\right\Vert .
$$
My efforts
I tried two different ways to prove that, both unsuccessfull.. 
First:
First, by squaring the whole inequality:
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)^{2}\left|\left\langle x,y\right\rangle \right|^{2}\leq\left\Vert x+y\right\Vert ^{2}\cdot\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}.
$$
We have from Cauchy-Schwarz that
$$
\left|\left\langle x,y\right\rangle \right|\leq\left\Vert x\right\Vert \cdot\left\Vert y\right\Vert 
$$
So we obtain 
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)^{2}\left|\left\langle x,y\right\rangle \right|^{2}\leq\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)^{2}\cdot\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}=\left(\left\Vert x\right\Vert ^{2}+\left\Vert y\right\Vert ^{2}+2\left\Vert x\right\Vert \left\Vert y\right\Vert \right)\cdot\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}.
$$
By Pythagorean theorem, we obtain
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)^{2}\left|\left\langle x,y\right\rangle \right|^{2}\leq\left(\left\Vert x+y\right\Vert ^{2}+2\left\Vert x\right\Vert \left\Vert y\right\Vert \right)\cdot\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}.
$$
We're almost there, except an extra term very annoying:
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)^{2}\left|\left\langle x,y\right\rangle \right|^{2}\leq\left\Vert x+y\right\Vert ^{2}\cdot\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}+2\left\Vert x\right\Vert ^{3}\left\Vert y\right\Vert ^{3}.
$$
Second
I tried after to use only the Cauchy-Schwarz inequality, not squared:
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\left|\left\langle x,y\right\rangle \right|\leq\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\cdot\left\Vert x\right\Vert \left\Vert y\right\Vert .
$$
My question
Could you give me a hint/idea on how to solve this problem? which Lemma/Theorem should I use?
Thank you
Franck

REPLY [5 votes]: Then, let me remove this question from the dead list of "unanswered questions" by answering it.
The statement is false. A counter-example is as follows. Let $E$ be $\mathbb{R}$ itself, and the inner product be the ordinary multiplication of real numbers. Let $x = 1$ and $y = -1$. Then the left hand side is $(||x||+||y||) \cdot | \langle x, y \rangle| = (1 + 1) 1 = 2.$ The right hand side is $||x + y|| \cdot ||x|| \cdot ||y|| = ||0|| \cdot 1 \cdot 1 = 0$.