The Cauchy criterion is the condition: For any (strictly) positive real. But nothing changes - the two definitions of a Cauchy sequence, one with ' $\le \varepsilon$' and the other using ' $\lt \varepsilon$', are equivalent. where Q>0 denotes the set of all strictly positive rational numbers. If it is your preference, you can make ' $\le \varepsilon$' into ' $\lt \varepsilon$' in at least two ways. A convergent sequence in a metric space always has the Cauchy property, but depending on the underlying space, the Cauchy sequences may be convergent or not. Using logic, algebra and simple inequality rules, one can show that for $n \ge 3$, $x_n$ is between the prior two terms, $x_$ arbitrarily ' $\varepsilon$ small', the assertion that $x_n$ is a Cauchy sequence can be backed up by finding an $N$ so that $|x_m - x_n| \le \varepsilon$ for $m,n \ge N$. A sequence (xn)nN ( x n) n N with xn X x n X for all n N n N is a Cauchy sequence in X X if and only if for every > 0 > 0 there exists N N N N such that d(xn,xm) < d ( x n, x m) < for all n, m > N n, m > N.
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