Question: Define $L(f)=\sup\left\{\frac{d_F(f(x),f(y))}{d_E(x,y)}:x,y\in E\right\}$ to be the Lipschitz seminorm. Show that $\forall f,g$ Lipschitz, and $\forall \lambda \in \mathbb{R}$ $L(f+\lambda g)\leq L(f)+|\lambda|L(g)$.
My Work: The only ting I can see is that since both $L(f)$ and $L(g)$ are going to be bounded above, namely by $K_1$ and $K_2$ respectively, then you can use the fact that $\sup(A+B)=\sup A +\sup B$ and so I find it to be an equality. Is this wrong? I believe it is correct but any comments would be great. Just a friendly reminder that this is homework.