Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 2E (Numbers and Sets)

The Fibonacci recurrence givesThe map $x\mapsto 1+1/x$ is strictly decreasing for $x>0$. Since $a_3=2\ge a_1=1$, applying this decreasing map reverses each inequality and proves by mathematical induction thatTaking even $n$ shows $a_{2n+2}\le a_{2n}$, so $(a_{2n})$ is a decreasing sequence.

The stated formula is Cassini's identity. It holds for $n=1$, and the recurrence giveswhich is the negative of the expression at index $n$. This proves the identity by induction.

ConsequentlyThe positive Fibonacci numbers tend to infinity, so this difference tends to zero.

Part (i) also shows that the odd subsequence is increasing and the even subsequence is decreasing. Each odd term is below the adjacent even term, so both are bounded monotone sequences and therefore have limits. Part (ii) says their difference tends to zero, so those limits coincide. Interlacing the two subsequences proves that $(a_n)$ has that common limit.