The Cauchy Condition states that a sequence of functions converges uniformly if, for every $\\epsilon > 0$, there exists a natural number $N$ such that for all $m, n \\geq N$, the supremum of the absolute difference between the functions is less than $\\epsilon$. This condition helps in establishing uniform convergence by ensuring that the functions get uniformly close to each other as the sequence progresses, which is essential when dealing with limits and continuity of function sequences.
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