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    Prove that U is a Subspace of V and is Contained in W

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    Let F be the field of real numbers and let V be the set of all sequences:

    ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise. Then V is a vector space over F.

    Let W = {( a_1, a_2, ... , a_n, ... ) belongs to V such that lim a_n = 0 where n tends to infinity }. Then W is a subspace of V.

    Let U = {( a_1, a_2, ... , a_n, ... ) belongs to V such that summation (a_i)^2 is finite}

    Prove that U is a subspace of V and is contained in W,
    where W = {( a_1, a_2, ... , a_n, ... ) belongs to V such that lim a_n = 0 where n tends to infinity }.

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    https://brainmass.com/math/vector-calculus/prove-subspace-contained-138360

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    Let be the field of real numbers and let be the set of all sequences , , where equality,
    addition and scalar multiplication are defined component wise. Then is a vector space over .
    Let . Then is a subspace of .
    Let .
    Prove that is a subspace of and is contained in , where .

    Solution:- Let .
    We have to prove that is a subspace of .
    Let and let such that
    where is finite and where is finite.
    Then

    where
    For
    ...

    Solution Summary

    This solution is comprised of a detailed, step by step response which illustrates how to solve this complex math problem related to subsets. The complete solution is provided in an attached Word file.

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