What is the Cauchy condition for a sequence?

The Cauchy condition states that for every positive tolerance

\varepsilon

, there exists an index

N

such that for all indices

m, n > N

, the distance between the sequence terms

|a_m - a_n|

is strictly less than

\varepsilon

. This means that all sufficiently late terms in the sequence become arbitrarily close to each other, regardless of how far apart their indices are.

Conditions

Fix an arbitrary positive tolerance $\varepsilon > 0$ .
Find an integer $N$ such that the tail of the sequence beyond $N$ is tightly clustered.
Verify that for every pair of indices $m, n > N$ , the absolute difference $|a_m - a_n| < \varepsilon$ .
Conclude that the sequence satisfies the Cauchy condition if this holds for all possible choices of $\varepsilon$ .

The script states: 'For every $ε>0$ , find N such that every pair m, $n>N$ satisfies |aₘ−aₙ|<ε.'

Believing that the Cauchy condition only requires consecutive terms to get close (e.g., $|a_{n+1} - a_n| \to 0$ ).
Thinking that $N$ must be the same for all tolerances $\varepsilon$ ; in reality, $N$ depends on $\varepsilon$ .
Confusing the Cauchy condition with the standard $\varepsilon$ - $N$ definition of a limit, which measures distance to a specific limit value rather than distance between pairs of terms.