(1) If $X$ is a some subset in $\mathbb{R}^n$,
$N=1$ : For $x\in X$, $1x\in {\rm conv}\ X$ so that $$ X\subset {\rm conv}\ X $$
$N=2$ : For $x,\ y\in X$, $$ \lambda_1x+\lambda_2 y \in {\rm conv}\ X,\ \lambda_1+ \lambda_2=1,\ \lambda_i\geq 0 $$ That is any line between points in $X$ is in ${\rm conv}\ X$.
(2) And $Y:={\rm conv}\ X$ is convex : If $x,\ y\in Y$, then $$ x= \sum_{i=1}^k\lambda_i x_i,\ y=\sum_{j=1}^{l} \tau_j y_i,\ x_i,\ y_j\in X $$
Then for $a,\ b\geq 0,\ a+b=1$, then we have $$ ax+by = \sum_{i=1}^ka\lambda_i x_i + \sum_{j=1}^{l} b\tau_j y_i $$ And note that $$
\sum_{i=1}^ka\lambda_i + \sum_{j=1}^{l} b \tau_j =1 $$
(3) $Y$ is smallest set which is a convex set containing $X$ : If $Z$ is a convex set containing $X$, then any line between points in $X$ is in $Z$. And if $$\lambda_i\geq 0,\ \sum_{i=1}^3\lambda_i=1,\ x_i\in X $$ then $$ x:=\sum_{i=1}^3\lambda_i x_i=
\lambda_1 x_1 + (1-\lambda_1) \sum_{i=2}^3 \frac{\lambda_i}{ 1-\lambda_1 } x_i $$ Note that $$ \sum_{i=2}^3 \frac{\lambda_i}{ 1-\lambda_1 } =1 $$
So $ \sum_{i=2}^3 \frac{\lambda_i}{ 1-\lambda_1 } x_i \in Z$ so that $x\in Z$ Continuously we have $ \sum_{i=1}^k\lambda_i x_i\in Z$ so that ${\rm conv}\ X\subset Z$.
(4) By definition $$ {\rm conv}\ X\subset {\rm cone}\ X $$
Let $$ Z:= \{ x\in \mathbb{R}^n\mid tx \in {\rm conv}\ X,\ {\rm
some}\ t\geq 0\} \cup \{0\}$$ Then
$$ Z = {\rm cone}\ X$$
Proof : $Z\subset {\rm cone}\ X $ : $$ x\in Z\rightarrow tx \in {\rm
conv}\ X \rightarrow tx=\sum_i c_i x_i,\ \sum_i c_i =1
$$
so that $$ x= \sum_i \frac{c_i}{t} x_i \in {\rm cone}\ X $$
${\rm cone}\ X\subset Z $ : Let $x=\sum_i c_i x_i \in {\rm cone}\
X,\ c:=\sum_i c_i$. Then $$ \frac{x}{c} = \sum_i \frac{ c_i}{c} x_i
\in {\rm conv}\ X $$ Let $t= \frac{1}{c}$ so that $x\in Z $.
Best Answer
Let $(x,y) \in B$. Let $n$ be a positive integer such that $ n >|x|$ and $1+n^{2} >\frac 1 y $. Let $t=\frac {n-x} {2n}$. Then $0<t<1$. Now $(x,y)=t(-n,y)+(1-t)(n,y)$. Note that $(-n,y)$ and $(n,y)$ both belong to $A$. Hence any element of $B$ is a convex combination of two points of $A$.