An Application Kolmogorov's Three Series Theorem

Question

I want to prove the following question, which is found in this practice exam:

My attempt so far is as follows - I just can't show that the $\sum E(Y_i)$ converges.

Answer 1

Write $$0 = E X_i = E[ X_i 1_{|X_i| \leq 1}] + E[ X_i 1_{|X_i| > 1}] = E[Y_i] + E[ X_i 1_{|X_i| > 1}]. $$

Thus $$|E[Y_i]| \leq E[|X_i| 1_{|X_i| > 1}] \leq E[\psi(X_i)].$$