Space of sequences/Linear subspaces/Exercise

Let ${}\mathbb {R} ^{\mathbb {N} _{+}}$ be the real vector space of all sequences. Show that the following subsets are linear subspaces.

a) The set of the constant sequences.

b) The set ${}\mathbb {R} ^{(\mathbb {N} _{+})}$ of the sequences where only finitely many members are different from ${}0$.

c) The set ${}F$ of the sequences that are constant with the exception of finitely many members.

d) The set ${}E$ of the sequences that have only finitely many different values.

e) The set of all convergent sequences.

f) The set ${}N$ of all null sequences. What inclusions do hold between these linear subspaces?

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