Given the unit sphere in $\mathbb{R}^n$ show that the extreme points of the unit ball are exactly the points on the unit sphere.
The geometry is obvious: Take any two points $||x||,||y||<1$ "truly" inside the ball, then their connection must be "truly" inside the ball as well.
My problem is using that approach as the basis of my proof, i.e. forming a convex combination $\lambda x+(1-\lambda)y$ and showing $||\lambda x+(1-\lambda)y||<1$ leads me to a big root term mess.
Is there a better way? Am I missing some algebraic trick to allow me to work with the root?
Edit: Not a duplicate of questions regarding function spaces. This is $\mathbb{R}^n$.
Edit2: Now that I see the answer for $\mathbb{R}^n$ the connection to the linked proof is clear though...
Hint: $\|x+y\|=\|x\|+\|y\|$ implies that $x$ and $y$ are linearly dependent. This can be seen by squaring both sides and using the condition for equality in C-S inequality: $|\langle x , y \rangle|=\|x\|\|y\|$ implies that $x$ and $y$ are linearly dependent. But if two unit vectors are $x$ and $y$ are linearly dependent then either $x=y$ or $x=-y$. Can you proceed?