Using induction prove that 1+1/4+1/9+...+1/n^2 < 2 -1/n for all positive integer n greater than 1.?

1 + 1/4 + 1/9 + ... + 1/n^2 < 2 - 1/n, where n is an arbitrary positive integer greater than 1.

If n = 2, then
1/1^2 + 1/2^2 = 1 + 1/4 = 5/4
2 - 1/2 = 3/2 = 6/4 > 5/4.
Thus, the inequality is true when n = 2.

Suppose the inequality is true when n = k, where k is a specific positive integer greater than 1.
If n = k + 1, then
1 + 1/4 + 1/9 + ... + 1/k^2 + 1/(k + 1)^2 < 2 - 1/k + 1/(k + 1)^2, and

2 - 1/k + 1/(k + 1)^2
= 2 - (k^2 + k + 1)/[k(k + 1)^2]
= 2 - {1/(k + 1) + 1/[k(k + 1)^2]}
= 2 - 1/(k + 1) - 1/[k(k + 1)^2].

Thus, 1 + 1/4 + 1/9 + ... + 1/k^2 + 1/(k + 1)^2 < 2 - 1/(k + 1) - 1/[k(k + 1)^2], or
{1 + 1/4 + 1/9 + ... + 1/k^2 + 1/(k + 1)^2} + 1/[k(k + 1)^2] < 2 - 1/(k + 1).

As k is positive,
1/[k(k + 1)^2] is greater than zero, and
{1 + 1/4 + 1/9 + ... + 1/k^2 + 1/(k + 1)^2} < {1 + 1/4 + 1/9 + ... + 1/k^2 + 1/(k + 1)^2} + 1/[k(k + 1)^2].
Certainly, {1 + 1/4 + 1/9 + ... + 1/k^2 + 1/(k + 1)^2} < 2 - 1/(k + 1).

Therefore, the inequality is true for all positive integer n greater than 1, by induction.