International Mathematical Olympiad (IMO) $2019$ , Day $1$ , Problem $3$ of $6$

Question:

A social network has $2019$ users, some pairs of whom are friends. Whenever User A is friends with User B, User B is also friends with User A. Events of the following kind may happen repeatedly, one at a time:

Three users A, B, and C such that A is friends with both B and C, but B and C are not friends, change their friendship statuses such that B and C are now friends, but A is no longer friends with B, and no longer friends with C. All other friendship statuses are unchanged.

Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Country that gave the Question: Croatia

The person that answers correctly and gives the official solution first - there is $2$ , go to International Mathematical Olympiad (IMO) Hall of Fame

Note - For this question, enter $1$ if it's correct (i.e. it can be proved) or $0$ if it's incorrect (i.e. it cannot be proved).