Infinite-dimensional space I've been dealing with an exercise asking to show that the infinite-dimensional space $R^\infty$ of infinite sequences is isomorphic to a proper subspace of itself. At first I thought I had to show that $R^\infty$ is isomorphic to any proper subspace of itself. But then it turned out that it was enough to present an instance of such a proper subspace, e.g. the subspace comprised by $(0,v_1,v_2,…)$, for which the isomorphism is easy to establish. So the former was a slight misinterpretation of the statement on my part. However:
Question: Is it actually at all a meaningful idea to ask oneself whether $R^\infty$ is isomorphic to any, i.e. an arbitrary, proper subspace of itself? If so, is it actually true or false?  

Two vector spaces on the same field are isomorphic iff have the same dimension. Interesting question for you: The (infinite-dimenional) subspace of eventually zero sequences is isomorphic to the whole space? 
