Put n=6 in 4n<(n^2-7): 24<29 is true. Now let n=6+1: 4(6+1)<((6+1)^2-7); 24+4<(36+12+1-7); 24+4<29+13. We know that the inequality was true for n=7, 24<29 so 24-29<0. If we subtract 29 from both sides of 24+4<29+13 we get (24-29)+4<13. But 24-29<0 so the left-hand side must be less than 4. Since 4 is less than 13 then any number less than 4 is also less than 13. The inequality remains true for n=7. By induction we can see that as n increases, the difference between the left and right hand expressions increases, with the right hand side increasing faster than the left. In other words, if n=6+r we have 24+4r<(36+12r+r^2-7); (24-29)+4r<12r+r^2. The expression on the left is less than 4r and expression on the right is greater than 12r, so left<right.