# linear algebra – Inequality for dimensions of eigenspaces

I want to proof that for an endomorphism $$f$$ over vector space $$V$$ with eigenvalue $$lambda$$ and
$$V = U oplus V/U$$ with $$f(U) subseteq U$$ the following holds:
$$dim E(f,lambda) leq dim E(f_U,lambda) + dim E(f_{V/U}, lambda)$$
I managed to proof that the inequality holds for the algebraic multiplicity of $$lambda$$ via splitting the characteristic polynomial of $$f$$. However I can’t find a way to connect that to the geometric multiplicity. I would appreciate any hint.