If \(p_{1}x_{1}+p_{2}x_{2}=y\) and \(x_{1}>0\) then it is feasible to decrease \(x_{1}\), so it must not be worth it for the consumer, implying that \(\text{MR}S_{2,1}\geq\frac{p_{2}}{p_{1}}.\)
If \(p_{1}x_{1}+p_{2}x_{2}<y\) and \(x_{1}>0,\ x_{2}>0\) then this means that for any sufficiently small \(\epsilon>0\) the ball or radius \(\epsilon\) around \((x_{1},x_{2})\) has all of its points being feasible and yet none of the points is strictly preferred to \((x_{1},x_{2})\), thus violating local non-satiation.