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The standard convergence definition requires knowing the limit $a$ in advance because it measures the distance $|a-a_n|$ to a specific number. The Cauchy condition avoids this limitation by measuring the pairwise distance $|a_n-a_m|$ between sequence terms, allowing one to verify convergence-like behavior without needing to know the limit beforehand.

Conditions: The sequence is being tested for convergence-like properties.; The candidate limit $a$ is unknown or not provided in advance.

The standard convergence definition requires knowing the limit $a$ in advance because it measures the distance $|a-a_n|$ to a specific number. The Cauchy condition avoids this limitation by measuring the pairwise distance $|a_n-a_m|$ between sequence terms, allowing one to verify convergence-like behavior without needing to know the limit beforehand.

Conditions: The sequence is being tested for convergence-like properties.; The candidate limit $a$ is unknown or not provided in advance.