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Prove that the sum of two subspaces of a vector space $V$ is a subset of $V$.

I am only asking to make sure that the solution to an exercise in my linear algebra book is logically correct in using a previously mentioned theorem to prove that the sum of two subspaces of a vector space $V$ is a subspace of $V$ that contains those two subspaces. If the sum of these two subspaces is not a subset of $V$, then it is obviously impossible for this sum to be a subspace of $V$. I just wanted to see a formal proof of this. Thank you.

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  • $\begingroup$ It is a quite obvious claim to prove. What is your attempt? $\endgroup$
    – user530422
    Commented Nov 3, 2018 at 17:33
  • $\begingroup$ I asked before attempting, but now I see that it is obvious. $\endgroup$ Commented Nov 3, 2018 at 17:42

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Let $u=a+b \in S_1+S_2$. Since each $S_i \subseteq V$, therefore $a,b \in V$. By closure under addition for $V$, $a+b \in V$. So $S_1+S_2 \subseteq V$.

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