Proving subsets are subspaces

Question

Let $V$ be the vector space of the sequences whose values are contained in $\mathbb R$. Prove whether or not the following subsets $W$ $\subseteq$ $V$ are subspaces of $(V,+, \cdot)$:

a) $W=$ {($a_n$) $\in$ $V$: $\sum_{n=1}^\infty$ |$a_n$| < $\infty$

b) $W=$ {($a_n$) $\in$ $V$: $\lim_{n\to \infty}$ $a_n$ = $1$

c) $W=$ {($a_n$) $\in$ $V$: $\exists$ $\lim_{n\to \infty}$ $a_{2n}$

d) $W=$ {($a_n$) $\in$ $V$: # {$n: a_n \neq 0$} < $\infty$ }

I know that to prove a given subset is a subspace requires me to prove that it is closed under vector addition and multiplication, and that the zero vector belongs to the subset. However, I have not yet studied Calculus and am unfamiliar with sequences, so I am a little lost about how to go about proving this.

Any help is much appreciated!

Answer 1

HINT

a) $W=$ {($a_n$) $\in$ $V$: $\sum_{n=1}^\infty$ |$a_n$| < $\infty$

• let check directly that the three properties hold

b) $W=$ {($a_n$) $\in$ $V$: $\lim_{n\to \infty}$ $a_n$ = $1$

• let consider $a_n=1$ and $b_n=1$ (or $a_n=0$)

c) $W=$ {($a_n$) $\in$ $V$: $\exists$ $\lim_{n\to \infty}$ $a_{2n}$

• let check directly that the three properties hold

d) $W=$ {($a_n$) $\in$ $V$: # {$n: a_n \neq 0$} < $\infty$ }

• let consider $a_n=0$