We have that for $x,y \in R^n,x^Ty \le 1, |y|_2 = 1$, can it be true that for some $i, x_i > 1$ and is it true that $|x|_2 \le 1$ ($|*|_2$-is the euclidian norm)?
$x_1y_1+x_2y_2+...+x_ny_n \le 1 = y_1^2+y_2^2+...+y_n^2$ I tried to get a contradiction assuming that $|x|_2 = x_1^2+x_2^2+...+x_n^2 \ge 1$. So, $y_1(x_1-y_1)+...+y_n(x_n-y_n) \le 0, x_1(x_1-y_1)+...+x_n(x_n-y_n) \ge 0$. However, I am stuck.
I also tried to do it with inner product and get $<x,y> = <x-y,y>+<y,y>$, since $<y,y> = 1, <x,y> \le 1$, then $<x-y,y> = 0$ so $x=y$.
Geometrically speaking, $|x^T y|$ only controls the length of the projection of $x$ in the direction of $y$. The length of $x$ itself can still be large. In fact, if $x\perp y$ and $x\neq 0$ (such a vector exists if $n\geq 2$), then $x^T y=0$, but $|x|_2$ can be made arbitrarily big by multiplying $x$ with a constant factor. Similarly, $x_i$ can be made arbitrarily big for some $i$.