$K$ cone, $x\in K$ and $y\in E$. Does $x+\lambda y\in K$ for all $\lambda\geq 0 $ implies $y\in K$?

Question

Suppose that $K\subset E$, where $E$ is a Banach space and $K$ is a closed convex cone. Fix $x\in K$ and $y\in E$. Assume that $x+\lambda y\in K$ for all $\lambda\geq 0$. Can we conclude that $y\in K$?

Thank you

Answer 1

Yes. Note that for $\lambda>0$, the condition that $x+ \lambda y \in K$ is equivalent to $\frac{1}{\lambda}x + y \in K$. But now as $\lambda \rightarrow \infty$, we have $(\frac{1}{\lambda}x + y) \rightarrow y$, so since $K$ is closed, the limit point $y$ must be in $K$.