How to show $\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$? Let $x,y \in \mathbb{R}^n$.
How can I show the following 
$$\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$$
The above has been used by the authors of the following paper on page 8, in first line
Online Principal Component Analysis.
Also, I think using the above for $M , N \in \mathbb{R}^{d \times k}$ the following is true.
$$\|M-N\|_F^2 \leq \|M\|_F^2+2|\text{tr}(M^TN)|$$
where $\|\cdot\|_F$ is Frobenius norm.

You'll have hard time to prove:
$$\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$$
Take $x = 0$ and $y \neq 0$, you get:
$$\|y\|_2^2 \le 0$$
which is unlikely.
