I somehow cannot prove this, so requesting assistance:
In the voting method known as the Nanson's method, the Borda counts of all candidates are computed. Next, those candidates with below average Borda counts (meaning a Borda count less than $(v(c+1))/2$, where $v$ is the number of voters and $c$ is the number of candidates) are eliminated. After the first round, the original preference rankings are used to construct a new set of preference rankings for the remaining candidates only. These preference rankings are then used to compute new Borda counts, those with below average counts are eliminated, and the process continues until a single candidate is left.
Show that Nanson's method will select the Condorcet winner if there is one by relating the different head-to-head vote counts to the Borda count.
Here is some further information:
Borda's method:
With Borda’s method, voters rank the entire list of candidates or choices in order of preference from the first choice to the last choice. After all votes have been cast, they are tallied as follows: On a particular ballot, the lowest ranking candidate is given 1 point, the second lowest is given 2 points, and so on, the top candidate receiving points equal to the number of candidates. The number of points given to each candidate and is summed across all ballots. This is called the Borda Count for the candidate. The winner is the candidate with the highest Borda count.
Condorcet winner:
First, for each pair of candidates determine which candidate is preferred by the most voters. If there is a candidate who wins EVERY comparison with all other candidates, then this candidate is the winner.
Thank you for your help.