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

[peregrintkanin]

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.

 

[fresh_42]

What are the properties that must be fulfilled in order for $S$ to be a subspace?

 

[PeroK]

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?