calculus - limit of the sequence $a_n=1+frac{1}{a_{n-1}}$ and $a_1=1$

Problem: Find with proof limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ with $a_1=1$ or show that the limit does not exist.

My attempt:

I have failed to determine the existence. However if the limit exists then it is easy to find it.

The sequence is not monotonic and I have failed to find any monotonic subsequence subsequence. The sequence is clearly bounded below by $1$. I have observed that the sequence is a continued fraction so it alternatively increases and decreases.

So, please help me.

Answer

As you already stated, the sequence alternately increases and decreases. Therefore you have found a monotone subsequence, namely $a_1,a_3,a_5,\dots$. And you have another monotone subsequence, namely $a_2,a_4,a_6,\dots$. You can easily find the recursion formula $a_n = f(a_{n-2})$.

Now prove that (a) both of these subsequences are indeed monotone and bounded and (b) that their limits are the same.

Alternative: Prove that $|a_j - a_{j-1}|$ is bounded by a geometric sequence, which will prove that the sequence in question is Cauchy.