Jordan decomposition of a restriction to a stable sub space

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having a question regarding the following problem:

f is an endomorphism from V to V, and its Jordan-Chevalley-Decomposition is f=h+n where h is semisimple and n is nil-potent.

No we have an f-stable subspace W of V, I am pretty sure that W is stable under h and n too, I can't seem to find a prove and I feel like I am missing something obvious.

Any help would be appreciated :)