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No, the observed success proportion does not necessarily get closer to the theoretical probability $p$ at every single step. The law of large numbers describes a long-run tendency where the proportion approaches $p$ as the number of trials grows, but it explicitly states that it need not get closer at every single step.

Conditions: The trials are independent.; Each trial has a common success probability $p$.; Analyzing the behavior of the proportion step-by-step.

No, the observed success proportion does not necessarily get closer to the theoretical probability $p$ at every single step. The law of large numbers describes a long-run tendency where the proportion approaches $p$ as the number of trials grows, but it explicitly states that it need not get closer at every single step.

Conditions: The trials are independent.; Each trial has a common success probability $p$.; Analyzing the behavior of the proportion step-by-step.