iterative sequence problem?

Question

I'm struggling on this but i get the feeling it's easier than i think.
The sequence is defined by:

x_n+1 = (x_n^5 + 2) / 6
where the initial value is 0.

and you need to show that

| x_n+1 - x_n |
after having shown that 0< x_n <1 (which i was able to do).

( by the way
can anyone help?

Answer 1

| x_(n+1) - x_n | = | [(x_n)^5 + 2] / 6 - [(x_{n-1})^5 + 2) / 6 |
= | (x_n)^5 - (x_{n-1})^5 | / 6
= | [(x_n) - x_(n-1)][(x_n)^4 + (x_n)³ x_(n-1) + (x_n)² (x_{n-1})² + (x_n)(x_{n-1})³ + (x_{n-1})^4] | / 6
≤ [(x_n) - x_(n-1)] (1+1+1+1+1) | / 6 (using x_m < 1 for all m)
= (5/6) | (x_n) - x_(n-1) |

Use this result and induction to get the second inequality

Answer 2

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