The aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal argument. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logics attempt to find a natural extension of traditional logic truth tables: the results they define are derived through probabilistic expressions instead. A difficulty with probabilistic logics is that they tend to multiply the computational c...
We have $a_n<b_n$ by the harmonic-arithmetic mean inequality.
Therefore $a_n<a_{n+1}$. It follows $b_{n+1}-a_{n+1} < \frac{a_n+b_n}{2}-a_n=\frac{b_n-a_n}{2}$. The proof follows.
