The Adventures of Ozzie and Harriet (1952)

S1E16 Stop Worrying

Ozzie notices that his family is preoccupied with their various anxieties, and he tries to reassure them by pointing out the absurdity of their worries. However, after coming across a magazine article, he finds himself struggling to manage his own feelings of anxiety, realizing that it's not as easy to dismiss worries as he thought.