Basketball player Michael keal's team statistician keeps track of the number, S(N), of the successful free throws he has made in the first N attempt of the season. Early in the season, S(N) was less than 80% of the N, but by the end of the season S(N) was more than 80% of the N. Was there necessarily a moment in between when S(N) was exactly 80% of N?
Use proof by contradiction and assume there was no such moments. Then S(N) much jump over 80% at one time....

2. Originally Posted by Vedicmaths

Basketball player Michael keal's team statistician keeps track of the number, S(N), of the successful free throws he has made in the first N attempt of the season. Early in the season, S(N) was less than 80% of the N, but by the end of the season S(N) was more than 80% of the N. Was there necessarily a moment in between when S(N) was exactly 80% of N?
Use proof by contradiction and assume there was no such moments. Then S(N) much jump over 80% at one time....
If there is no such moment then some time in the season there exists a number of free throws $N$ where the number of successes was $M$ and

$\frac{M}{N}<0.8$

but the next free throw was a success and:

$\frac{M+1}{N+1}>0.8$

The first of these inequalities may be rearranges to:

$M-0.8N<0$

and thesecond to:

$M-0.8N>0.8$,

but $M-0.8N$ canot be both less than $0$ and greater than $0.8$, a contradiction, hence the assumption is false and there must be a point in the season where $S(N)=80\%$

RonL