Consider a subspace $H$, where $H= \{(x_1, x_2, \dots)\in \mathbb{F} \ | \ x_1 = 0 \} $ is a subspace of the "sequence room" $\mathbb{F}$ (the room $\mathbb{F}$ defined to contain all real sequences of the type $\mathbf{x}= \{x_j\}^\infty_{j=x}=(x_1,x_2,x_3,\dots )$ ).
In my notes I have as an example that, if $x_1 = 2$, then $H$ is not a subspace of $\mathbb{F}$. Why is that?
Is it simply because, with $x_1$ defined as nonzero, then the zero vector can not be in $H$? Hence $(0,0,0,\dots) \notin H$, and thus one of the criteria for a subspace is not satisfied?
Is that it, or is there something else I have missed?