real analysis - Evaluating the limit of a sequence given by recurrence relation $a_1=sqrt2$, $a_{n+1}=sqrt{2+a_n}$. Is my solution correct?

Problem

The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:

• $a_1=\sqrt2$,


• $a_{n+1}=\sqrt{2+a_n}$.

Evaluate the limit $\lim_{n\to\infty} a_n$.

Solution

• Show that the sequence $(a_n)_{n=1}^\infty$ is monotonic. The statement $$V(n): a_n < a_{n+1}$$ holds for $n = 1$, that is $\sqrt2 < \sqrt{2+\sqrt2}$. Let us assume the statement holds for $n$ and show that $V(n) \implies V(n+1)$. We have that $$a_n < a_{n+1}.$$ Adding 2 to both sides and taking square roots, we have that $$\sqrt{2+a_n} < \sqrt{2+a_{n+1}},$$ that is $a_{n+1} < a_{n+2}$ by definition.


• Find bounds for $a_n$. The statement $$W(n): 0 < a_n < 2$$ holds for $n=1$, that is $0 < \sqrt2 < 2$. Let us assume the statement holds for $n$ and show that $W(n) \implies W(n+1)$. We have that $$0 < a_n < 2.$$ Adding two and taking square roots, we have that $$0 < \sqrt2 < \sqrt{2+a_n} < \sqrt4 = 2.$$


• The limit $\lim_{n\to\infty} a_n$ exists, because $(a_n)_{n=1}^\infty$ is a bounded monotonic sequence. Let $A = \lim_{n\to\infty} a_n$.


• Therefore the limit $\lim_{n \to\infty} a_{n+1}$ exists as well and $\lim_{n \to\infty} a_{n+1} = A$. (For $(n_k)_{k=1}^\infty = (2,3,4, \dots)$, we have that $(a_{n_k})_{k=1}^\infty$ is a subsequence of $(a_n)_{n=1}^\infty$, from which the statement follows.)


• We have that $a_{n+1} = f(a_n)$. That means that $A = \lim_{n\to\infty} a_n = \lim_{n \to\infty} {f(a_n)} = f(\lim_{n \to\infty} a_n) = f(A) = \sqrt{2 + A}$. Solving the equation $A = \sqrt{2 + A}$, we get $A = -1 \lor A = 2$.


• Putting it all together, we get that $A = 2$, because the terms of the sequence are increasing and $a_1 > 0$.

Is my solution correct?

Answer

Looks great. Here is a fun trick I've seen to answer this question.

Using the half angle formula, notice the following:

$$\cos\left(\frac{\pi}{4}\right)=\frac{1}{2}\sqrt 2\\\cos\left(\frac{\pi}{8}\right)=\sqrt{\frac{1}{2}(1+\frac{1}{2}\sqrt 2)}=\frac{1}{2}\sqrt{2+\sqrt 2}\\\cos\left(\frac{\pi}{16}\right)=\sqrt{\frac{1}{2}(1+\frac{1}{2}\sqrt{ 2+\sqrt2})}=\frac{1}{2}\sqrt{2+\sqrt {2+\sqrt 2}}\\\vdots\\\cos\left(\frac{\pi}{2^{n+1}}\right)=\underbrace{\frac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2+{\ldots}}}}}_\text{n times}=\frac{1}{2}a_{n}$$

Now let $n$ approach infinity.