Space of sequences/Linear subspaces/Residue class spaces/Exercise

We consider the real sequences

$x_n=\{\begin{array}{c}1,\text{ if }n\text{ even},\\ 0,\text{ else},\end{array}$

and

$y_n=\{\begin{array}{c}1,\text{ if }n\text{ odd},\\ 0,\text{ else},\end{array},$

and we use the notations from exercise.

a) Show that the two sequences $x_n$ and $y_n$ are linearly independent (considered) in ${ℝ}^{{\mathbb{N}}_{+}}/{ℝ}^{({\mathbb{N}}_{+})}$.

b) Show that the two sequences $x_n$ and $y_n$ are linearly dependent in ${ℝ}^{{\mathbb{N}}_{+}}/F$.

c) How is it in ${ℝ}^{{\mathbb{N}}_{+}}/N$?

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