Question

Show that the sum of any positive real number and it's reciprocal cannot be less than 2

Answer 1

Since we are dealing with positive real numbers, and the sum resulting from n + 1/n is the same for all n >= 1 as it is for n <= 1, begin the proof with the statement, let n >= 1

n + 1 >= 2

(n + 1)(n - 1) >= 2(n - 1)

n^2 - 1 >= 2n -2

n^2 + 1 >= 2n

n + 1/n >= 2

The justification for each step above is based on the property of real numbers.