solution set of $2x+y\ge{8},x+2y\ge 10,x\ge0,y\ge0$ 
How do I prove that the solution set of the inequalities $2x+y\ge{8},x+2y\ge 10,x\ge0,y\ge0$ is an unbounded region ?

I can see that the statement is true graphically by plotting each inequalities. But what does it mean mathematically ?
How do I prove it without actually plotting it ?
 A: It is unbounded because if you solve the system of inequalities which means that ALL of them must be satisfied you get the region $\mathcal{R}$ of the plane where
$(y\geq 8-2x) \land (y\geq 5-\dfrac{x}{2})\land (x\geq 0 )\land (y\geq 0)$
As you can see all the results contain $\geq$  which means that, even if we want to be cautious, we can take the largest values for $x$ and for $y$ which satisfy ALL the previous inequalities for instance $\mathcal{S}=(x\geq 10;\;y\geq 10)$  and we have a subregion of the plane $\mathcal{S}\subset \mathcal{R}$ which is unbounded, so to a greater extent is unbounded $\mathcal{R}$
Hope this helps
A: If you show there is no value for max/min then you prove that it is unbounded. Feasible region
A: Let $(x_n, y_n)=(10n, 10n)$; 
it is easy to chech out 
this point satisfies all the above inequalities.
Also notice that 
$ 
\underset{n \rightarrow }{\lim}x_n= 
\underset{n \rightarrow }{\lim}x_n= 
\infty $; so this region can not be bounded; 
so this region must be unbounded! 


Remark(I): 
For every point $(z,w)$ and for every real number $0 < M$; 
there exists $n$ such that: 
$$\sqrt{(x_n-z)^2+(y_n-w)^2}=|| (x_n,y)-(z_n,w) || > M$$ 

Proof of boundedness. 
Suppose  on contrary that: 
this region is bounded; 
so there exist a point $(z,w)$ and a radios $0 < M$ 
such that for every point (x,y) of the region 
we have: 
$$\sqrt{(x-z)^2+(y-w)^2}=|| (x,y)-(z,w) || \leq M.$$ 
But notice that for every $n \in \mathbb{N} $, 
$(x_n,y_n)$ belongs to this region; 
which has an obvious contradiction with Remark(I) 




Remark(II): 
Note that in a euclidian's plane 
if a region  is bounded 
then the sequence of 
the $x$-coordintate of 
every sequence must have a accumulation point.
