Please give me help on this question 5 10 points Prove that

Please give me help on this question.

5. (10 points) Prove that for every positive integer k, the following is true For every real number r 0, there are only finitely many solutions in positive integers to nk r. In other words, there exists some number m (that depends on k and r) such that there are at most m ways of choosing a positive integer n 1, and a (possibly different positive integer n2, etc., that satisfy the equation.

Solution

For every positive integer K, real number r > 0,

1/n1 + 1/n2 + 1/n3 + ......+1/nk = r

given k is positive integer so we can any value greater than zero

suppose take n1= 1 , n2= 2, n3 = 3,....n10 = 10

then the equation is (1/1) + (1/2) +(1/3) +(1/4)+.....+ (1/10) = 2.928

so from the above examples we conculed that for every positive interger of n values , the K is always a positive integer.