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## Homework Statement

Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges.

## Homework Equations

## The Attempt at a Solution

First, we must prove that the sequence is bounded below. We will prove that it is bounded below by 0. ##t_1 = 1 \ge 0##, so the base case holds. Now, suppose that ##t_k \ge 0##. Then ##t_{k+1} = (1 - \frac{1}{4k^2}) t_k \ge (1 - \frac{1}{4k^2}) (0) = 0##. So the induction is complete, and the sequence is bounded below by 0.

Now we must show that the sequence is decreasing. ##t_2 = 15/16 \le 1 = t_1##, so the base case holds. Now, suppose that ##t_{k+1} \le t_k##. Then ##t_{k+2} = (1 - \frac{1}{4(k+1)^2})t_{k+1} \le t_{k+1}##. So the induction is complete, and the sequence is decreasing.

By the Monotone Convergence Theorem, the sequence converges.