Law Of Large Numbers, Central Limit Theorem, Normal Distribution, Random Variable, Jacob Bernoulli.
L.H.S. is the CDF of Z_n while that on the R.H.S is the CDF of N(0, 1). In (b) and (c), P(Z_n ≤ z) is just another way to specify the CDF of Z_n. Φ(z) is the notation customarily used for the CDF of the standard normal random variable N(0, 1). In (c), the wavy equals (or half-equals ‘ ≃’) means