# Prove that W is a Subspace of V

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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 | lim n -> infinity a_n = 0}.

Prove that W is a subspace of V.

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This solution is comprised of a detailed explanation of sub spaces of a vector space. It contains a step-by-step explanation for the problem. Notes are also given at the end and the solution is provided in a Word document.

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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 .

Prove that is a subspace of .

Solution: Let .

Let and let such that

where and where .

Then

...

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- BSc, Manipur University
- MSc, Kanpur University

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