Let $W$ be a subspace of a vector space $V$ .Show that $W \subseteq \textbf{v} + W \Rightarrow \textbf{v} + W \subseteq W.$

Question

Show that $W \subseteq \textbf{v} + W \Rightarrow \textbf{v} + W \subseteq W.$

Guys, i have edited my original solution in the link below. Now here is my proof for the converse. Main thing is, is my proof now ok? Although i did not start straight away with let $x \in v+W$, then show $x \in W$, but i felt that my first part is necessary.

Note that for any $ \mathbf{w} \in W$ , we have $\mathbf{w} \in \mathbf{v} + W$ as well. Then $$ \mathbf{w} = \mathbf{v} + \mathbf{w}_{1} ~\text{for some } \mathbf{w_1} \in W ~~~ \implies ~~~ \mathbf{v} = \mathbf{w} - \mathbf{w}_{1} \in W$$ Let $$ \mathbf{x} \in \mathbf{v} + W, ~~\mathbf{x} = \mathbf{v} + \mathbf{w}_{2} = \mathbf{w} - \mathbf{w}_{1} + \mathbf{w}_{2} \in W$$ Thus, $\mathbf{v} + W \subseteq W$.

Let $W$ be a subspace of a vector space $V$ . Show that the following are equivalent.

Answer 1

It seems correct, but more complicated than needed.

Suppose $W\subseteq\mathbf{v}+W$. Then $\mathbf{0}=\mathbf{v}+\mathbf{w}$, for some $\mathbf{w}\in W$. Therefore $$ \mathbf{v}=-\mathbf{w}\in W $$ and so $\mathbf{v}+W\subseteq W$.