Prove by mathematical induction that the summation of 1/(i^2), from i=1 to n, is (< or =) 2 - (1/n)

I started working with it (using n+1), but the algebra just doesn't seem to be coming together. Assistance would be greatly appreciated.

Base Case (n = 1):
1/1^2 = 1 = 2 - 1/1.

Inductive Step:
Assuming that 1 + 1/2^2 + ... + 1/k^2 ≤ 2 - 1/k:

1 + 1/2^2 + ... + 1/k^2 + 1/(k+1)^2
= [1 + 1/2^2 + ... + 1/k^2] + 1/(k+1)^2
≤ (2 - 1/k) + 1/(k+1)^2, by inductive hypothesis

It suffices to show that
-1/k + 1/(k+1)^2 ≤ 1/(k+1) for all k > 0.

This is true
<==> -(k+1)^2 + k ≤ k(k+1), clearing fractions
<==> -(k+1)^2 ≤ k^2, which is clearly true for k > 0.

I hope this helps!