Prove this sequence is increasing

real-analysissequences-and-series

The sequence $A_n$ is described as-
\begin{align*}
A_1 & = \sqrt{2}\\
A_{n+1} & =(2+A_n)^{0.5}
\end{align*}
where $A_n$ is the $n$th term in the sequence.

I can prove by induction that this sequence is bounded by above by $2$ and then find the limit to be $2$. But in order to assume that the limit exists, I need to show that it is also increasing so I can use the monotone convergence theorem. It is easy to see why it is increasing, but how do I prove it?

Best Answer

We need to prove that

$$A_{n+1}=\sqrt{2+A_n}\ge A_n \iff A_n^2-A_n-2 \le 0 \iff -1\le A_n \le2$$

then we have $A_n \ge 0$ and by inducion we can prove that $A_n\le 2$ indeed

  • base case: $A_1=\sqrt 2\le 2$
  • induction step: assuming true $A_n\le 2$ we need to prove that $A_{n+1}\le 2$ then

$$A_{n+1}=\sqrt{2+A_n} \le\sqrt{2+2}=2$$

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