Addition of two L-smooth function is also L-smooth?

Question

Assume $f(x)$ has an L-Lipschitz continuous gradient say $L_1$ i.e there is a constant L>0 such that $$\|\nabla f(x) - \nabla f(y)\|_2 \le L\|x-y\|_2$$ for any $x,y$.

Also $g(x)$ has an L-Lipschitz continuous gradient, say $L_2$. Is $f(x)+g(x)$ has an L-Lipschitz continuous gradient?

I tried to use property of L2 norm but couldn't be sure if this is correct. Since $\| x + y\|_2 \leq \| x\|_2 + \| y\|_2$, the $L$ of $f(x)+g(x)$ would be $L_1+L_2$

I would appreciate any help.

Answer 1

I may be wrong but this seems like a simple use of sub-additivity of the norm and additivity of the gradient:

$||\nabla \big( f+g \big)(x) - \nabla \big(f+g)(y)||_2=\Big\Vert \nabla f(x)+\nabla g(x)- \big( \nabla f(y)+\nabla g(y) \big) \Big \Vert_2=$

$=\Big \Vert \big( \nabla f(x)-\nabla f(y) \big) + \big( \nabla g(x)-\nabla g(y) \big) \Big \Vert_2\leq \big \Vert \nabla f(x)-\nabla f(y) \big \Vert_2+ \big \Vert \nabla g(x)-\nabla g(y) \big \Vert_2\leq$

$\leq L_1\cdot \Vert x-y \Vert_2+ L_2 \cdot \Vert x-y \Vert_2= \big( L_1+L_2 \big) \cdot \Vert x-y \Vert_2 $