Ordered field/Sequences/Zero sequences as linear subspace/1 over n and 1 over n^2/Exercise

Let ${}K$ be an ordered field, and let

${}V=K^{\mathbb {N} _{+}}\,$

be the vector space of all sequences in ${}K$ (with componentwise addition and scalar multiplication).

a) Show that (without using theorems about convergent sequences), the set of zero sequences, that is,

${}U={\left\{(x_{n})_{n\in \mathbb {N} _{+}}\mid (x_{n})_{n\in \mathbb {N} _{+}}{\text{ converges to 0}}\right\}}\,$

is a ${}K$-linear subspace of ${}V$.

b) Are the sequences

$(1/n)_{n\in \mathbb {N} _{+}}{\text{ and }}(1/n^{2})_{n\in \mathbb {N} _{+}}$

linearly independent in ${}V$?

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