Final answer:
Iago offers 'proof' of Cassio and Desdemona's affair by claiming that he had heard Cassio talking in his sleep and witnessed Cassio wiping his beard with Desdemona's handkerchief.
Step-by-step explanation:
Iago offers 'proof' of Cassio and Desdemona's affair by claiming that he heard Cassio talking in his sleep and witnessed Cassio wipe his beard with Desdemona's handkerchief, which Iago had previously planted in Cassio's room. On top of that, Iago states that he saw Cassio and Desdemona talking privately, further supporting his accusation. These acts are central to Iago's plot to manipulate Othello into believing that his wife is unfaithful.