Reverse Triangle Inequality in vector space

I have already proved Triangle Inequality $\lVert v+w\rVert \le \lVert v \rVert+\lVert w \rVert$ in vector space $\mathbb{R}^n$. However, I'm having difficulties proving the Reverse Triangle Inequality in the vector space.

I have started with

$$\lVert v \rVert^2=v \cdot v-w\cdot w+w\cdot w \le \lVert v-w \rVert+\lVert w \rVert^2$$ $$\lVert w \rVert^2=w \cdot w-v\cdot v+v\cdot v \le \lVert w-v \rVert+\lVert v \rVert^2$$

based on the Triangle Inequality but I don't know if it is correct. If this is right, I think I can go on with proving it the same way I would the normal Reverse Triangle Inequality.

Solution 1:

$$\lVert v \rVert = \lVert v -w +w \rVert \le \lVert v -w \rVert + \lVert w \rVert$$ $$\lVert w \rVert = \lVert w -v +v \rVert \le \lVert w -v \rVert + \lVert v \rVert = \lVert v-w \rVert + \lVert v \rVert$$

Thus we see that $\lVert v-w \rVert \ge \Big|\lVert v \rVert - \lVert w \rVert\Big|$ as claimed.