What makes Algernon suspect Ernest’s identity in Oscar Wilde's The Importance of Being Earnest?

1.He overhears Ernest talking about his wife who lives in the country.
2.He finds a note with Ernest’s actual address in his coat pocket.
3.He receives a letter from a woman claiming to be Ernest’s niece.
4.He finds Ernest’s cigarette case inscribed with the name Jack.

At the beggining of the play Algernon, Jack´s best friend and cousin of Gwendolen Fairfax the love interest of Ernest, found an inscription in his cigarette case addressed to "Uncle Jack" from someone who refers herself as "Little Cecily".

Algernon suspects that Jack who he knows as Ernest is leading a double life and called him a Bunburyist after a non existent friend Algernon pretend to have.

Algernon had invented a chronic invalid characer named Bunbury to give him a suspension of his real life. He must be near his deathbed which conveniently draw him away from his social obligations which bored him. Unlike Jack who did not admit even to himself the fictionality of Ernest; Algernon admits his wrongdoing and revels on it.

Soon after he beggins to suspect on Jack´s double life he presented to his door and announced his intentions to marry Gwendolen. Algernon confronted him with his discovery and demands to know the truth, Jack confesses and tells him who Cecily was and without meaning to awoke Algernon´s attention toward his ward.

Algernon decides to put Bunbury aside and create another fictional character, he would be Jack´s brother Ernest to catch Cecily´s attention and heart.

I hope this answer is correct and help you. I give a little resume of the act I and part of the second one in order to not spoil the rest of the play if you have not read it yet. If you have any doubt don´t hesitate to ask. Greetings.