I must find the boundary, closure and interior of this set\begin{equation*}S = \{\mathbb{x}: x_1 + x_2 \leq 5, -x_1 + x_2 + x_3\leq 7,x_1,x_2,x_3\geq0 \}\end{equation*}For the closure, i plan to proof that the set is closed. But I feel that my reasoning is wrong, i take a convergent sequence in $S$ with the shape $\{x^{(n)} \}$ such that:\begin{equation*}x^{(n)} = ({x_1}^{(n)},{x_2}^{(n)},{x_3}^{(n)}) \to (x_1,x_2,x_3) = x\end{equation*}how the sequence is in the set:\begin{equation*}{x_1}^{(n)} + {x_2}^{(n)}\leq 5, -{x_1}^{(n)} + {x_2}^{(n)} + {x_3}^{(n)}\leq 7,{x_1}^{(n)},{x_2}^{(n)},{x_3}^{(n)} \geq 0\end{equation*}and taking the limit\begin{equation*}x_1 + x_2 \leq 5, -x_1 + x_2 + x_3\leq 7,x_1,x_2,x_3\geq0\end{equation*}This probably doesn't make sense but that's why I'm looking for help, I'm also very interested in the interior and the closure (it's best to use that $bs \ S = \overline{S}-int \ S $)
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