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**1. The problem statement, all variables and given/known data**

Prove the the union of three subspaces is a subspace if one of the subspaces contains the others

**2. Relevant equations**

A subset

**W**of a vector space V is called a subspace if : 1) ##0 \in W ##. 2) if ##U_1## and ##U_2## are in ##W##, then

##U_1 + U_2 \in W##, 3) if ##\alpha ## is a scalar, then ##\alpha U\in W##

**3. The attempt at a solution**

assume that ##\exists~x,y,z \in U_1\cup U_2\cup U_3 ~## such that, ##x \in U_1 ~, y \in U_2 ~ and~~ z \in U_3##.

We know that, ##x+y+z~\in U_1\cup U_2\cup U_3##, hence ##x+y+z~is~in~either~U_1 ~or~U_2 ~or ~U_3##

Assume, WOLOG, that ##x+y+z~\in~U_1 ,~then~ y+z \in U_1 ,~moreover,~y+z\in U_1 \cup U_2~##,thus

##z\in U_1 \cup U_2~,~and~we~have~two~cases~to~consider##.

##i)~ z \in U_1 ~,~then~y+z\in U_1 ,~\implies~y\in U_1 ~, thus,~ any~z\in~U_3 ~, then~z\in U_1~,~and~any~y \in~U_2 ~, then~y\in U_2##

##hence,~U_2 ~and~U_3 ~\subset U_1##

##ii) ~z\in U_2##, then I don't know how to proceed.

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