We don't have to use Lagrange or Cauchy , those are simple sequences.

Let's stay with your problem:

A) we will use mathematical induction:

for n=1 we have 1=<2=<3 so 1=<a_{0}=<3

we assume that it's true for n and let's prove it for n+1

since 1

3 and f is increasing we have

End of induction .

because a_n is between 3 and 1 the numerator is negative

and denominator is positive so the whole thing is negative

not positive

B) let's prove that (a_{n}) is increasing :

we have now proved that it's increasing .

Now and since it's bounded , it converges.

As i said , to compute the limit , solve the equation f(x)=x