Sequences convergence problem

Let a₁ > 1, and suppose a_n+1 = 2 - 1/a_n for all n >= 2 . Prove that (a_n) converges, and find its limit.

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Originally Posted by Kanwar245

Let a₁ > 1, and suppose a_n+1 = 2 - 1/a_n for all n >= 2 . Prove that (a_n) converges, and find its limit.

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It is sufficient to show that the sequence is decreasing and bounded below.
Do that with induction.

The solve the equation $\displaystyle L=2-\frac{1}{L}$.

Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2

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Originally Posted by Kanwar245

Don't you think there's a typo, since we are given a1 > 1, and a_n+1 is for n>= 2

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Why should anyone think that?

Take a look at this

Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that a_n+1 >= a_n right

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Originally Posted by Kanwar245

Okay, so when proving through induction that it's monotonically decreasing, I just need to prove that a_n+1 >= a_n right

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BASE CASE: Prove that $\displaystyle a_1>a_2\ge 1$.

PROVE
Inductive case: If $\displaystyle a_k>a_{k-1}\ge 1$ then $\displaystyle a_{k+1}>a_k\ge 1$.