convergence of sequence defined recursively $\frac{2}{a_{n+2}}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$

Note that $2\left(\frac{1}{a_{n+2}}-\frac{1}{a_{n+1}} \right)=-\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n}}\right)$, thus with $b_n = \frac{1}{a_{n+1}}-\frac{1}{a_{n}}$, $b_{n+1}=-\frac 12 b_n$.

Hence $b_{n+1} = \frac{(-1)^n}{2^n}b_1$, so $\sum_n b_n$ converges, thus $\frac1{a_n}$ converges, and so does $a_n$.

Let $\ell=\lim_n a_n$. Then $\sum_{n=1}^\infty b_n = \frac 1\ell -\frac 1{a_1}$, which rewrites as $\frac 23 \left(\frac 1{a_1} - \frac 1{a_2} \right) = \frac 1\ell -\frac 1{a_1}$, hence $$\ell = \frac{1}{\frac 23 \left(\frac 1{a_1} - \frac 1{a_2} \right)+\frac 1{a_1}}$$

As it was suggested in the comments use $b_n=\frac{1}{a_n}$ and then solve $$2b_{n+2}=b_{n+1}+b_n$$ using characteristic polynomials technique (or look at this similar question), leading to the following polynomial $$2x^2-x-1=0$$ with solutions $x_1=1$, $x_2=-\frac{1}{2}$ and general solution for the recurrence $$b_n=Ax_1^n+Bx_2^n=A+B\left(-\frac{1}{2}\right)^n \Rightarrow a_n=\frac{1}{A+B\left(-\frac{1}{2}\right)^n}$$ and $\lim\limits_{n\rightarrow\infty} a_n = \frac{1}{A}$. Given $a_1=\frac{1}{A-\frac{B}{2}}$ and $a_2=\frac{1}{A+\frac{B}{4}}$, it's not difficult to find $A$: $$A-\frac{B}{2}=\frac{1}{a_1} , 2A+\frac{B}{2}=\frac{2}{a_2} \Rightarrow A=\frac{1}{3a_1}+\frac{2}{3a_2}$$

Too involved for a comment: Another way to prove the limit exists:

For positive $a, b$ let $H(a,b)=(\frac {1}{2}(a^{-1}+b^{-1}))^{-1}.$ We have $a_{n+2}=H(a_{n+1},a_n).$

We have $0< a\leq b\implies a\leq H(a,b)\leq b.$ Therefore the sequence $S=(a_{2n-1})_{n\in \Bbb N}$ is increasing and the sequence $T=(a_{2n})_{n\in \Bbb N}$ is decreasing, and every term of $S$ is $\leq$ every term of $T$. So $S$ converges to a limit $D$ while $T$ converges to a limit $U,$ with $0<a_1\leq D\leq U\leq a_2.$

Observe that $H:\Bbb R^+\times \Bbb R^+\to \Bbb R^+$ is continuous, so $$D=\lim_{n\to \infty}a_{2n+1}=$$ $$=\lim_{n\to \infty}H(a_{2n}, a_{2n-1})=$$ $$=H(\lim_{n\to\infty}a_{2n},\lim_{n\to \infty}a_{2n-1})=H(U,D).$$ But for any positive $U,D$ we have $H(U,D)=D \iff U=D.$

2/a[n+2] = 1/a[n+1] + 1/a[n] , get common denominator

2/a[n+2] = [a(n+1) + a(n)]/[ a(n+1) . a(N) ] , invert

a[n+2] = 2[a(n+1).a(n)]/[ a(n+1) + a(n) ] eqn 1

a(2) = 2 a(1).a(0)/ [1a(1) +1a(0)] eqn 2 , note 1+1=2^1

a(3) = 2 a(2).a(1)/[ a(2) + a(1)] eqn 3 now substitute eqn 2 into eqn 3 and get

a(3) = 4a(1).a(0)/[1a(1) + 3a(0)] eqn 4 , note 1+3=4=2^2

using similar substitutions

a(4) = 8a(1).a(0)/[ 3a(1) +5a(0)] eqn 5 , and 3+5=8=2^3

a(5) = 16a(1).a(0)/[5a(1) +11a(0)] eqn 6, and 5+11=16=2^4

The sums of the co efficients of the denominator = the co efficient of the numerator and the co efficient of the previous a(0) becomes the co efficient of the next a(1) , so we will expect the next term

a(6) = 2^5.a(1).a(0)/[ 11a(1) +p.a(0)] where p= 2^5-11=21 , so

a(6) = 32 a(1).a(0) /[ 11a(1) + 21a(0)]

a(n) = 2^(n-1).a(1).a(0)/[ (2^(n-1)-p)a(1) +p a(0)]

At this stage divide all the terms of the numerator and the denominator by 2^(n-1) , and let n tend to infinity .

Then the limit of a(n) and 1/a(n) has a limit as n tends to infinity.