Let $V$ be an inner product space and $u,v,w\in V$. Choose the correct option

1. $|\langle u,v\rangle| \le || u|| + ||v||$
2. $|\langle u,v\rangle| \le \frac{1}{2}(|| u||^2 + ||v||^2)$
3. $|\langle u,v\rangle \le |\langle u,w \rangle | + |\langle w,v \rangle |$
4. $|| u +v|| \le ||u +w|| + ||w +v||$

**My attempt:**

For option $1)$ i take $u =(2,0)$ and $v= (3,0)$

now $|\langle u,v\rangle| = \langle (2,0), (3,0) \rangle = 6$ and $||u|| =2, ||v|| =3$ that is option $1$ is false

option $2$ is false take $u =(3,0)$ and $v =(0,3)$

option $3$ and $4 $ is true by triangle inequality

Is it correct?