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.
We don't have necessarily an equality, for example if $g=f$ and $\lambda=-1$. But we have, assuming $d_F$ comes from a norm, $$\small d(f(x)+\lambda g(x),f(y)+\lambda g(y))=\lVert f(x)-f(y)+\lambda(g(x)-g(y))\rVert\leq L(f)d(x,y)+|\lambda| L(g)d(x,y).$$