Prove that ##S## is a subspace of ##V##

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To determine if the subset ##S## of real sequences converging to zero is a subspace of the vector space ##V## of all real sequences, it must satisfy three properties: it must contain the zero vector, be closed under addition, and be closed under scalar multiplication. The zero sequence is included in ##S## since its limit is zero. Additionally, the sum of two sequences that converge to zero also converges to zero, and a scalar multiple of a sequence that converges to zero will also converge to zero. Thus, ##S## meets all criteria to be a subspace of ##V##. The discussion highlights the application of vector space theory to infinite-dimensional sequences.
peregrintkanin
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Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##?

Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied about vector spaces and vector subspaces.
 
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What are the properties that must be fulfilled in order for ##S## to be a subspace?
 
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peregrintkanin said:
Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##?

Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied about vector spaces and vector subspaces.
If ##(a_1, a_2, a_3)## represents a three-dimensional vector, then why can't an infinite sequence ##(a_1, a_2, a_3 \dots )## represent an infinite dimensional vector?
 

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