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## Homework Statement

Let F be the field of all real numbers and let V be the set of all sequences (a1,a2,...a_n,...), a_i in F, where equality, addition, and scalar multiplication are defined component-wise.

(a) Prove that V is a vector space over F

(b) Let W={(a1, a2,...,a_n,...) in V | lim a_n = 0 as n-->inf}. Prove that W is a subspace of V.

(c) Let U={(a1,...,a_n,...) in V | summation of (a_i)^2 is finite, i evaluated from 1 to inf}. Prove that U is a subspace of V and is contained in W.

## The Attempt at a Solution

I know that in order for W to be a subspace of V, W must form a vector space over F under the operations of V. I've already proved (a). Do I need to know the limit of a_n to prove (b) or is that just for (c)? It seems like proving (b) is pretty similar to (a), right? Any tips on proving (c)?