# Interactive Real Analysis - part of MathCS.org

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## Example 6.2.4(c):

If f(x) is continuous in a domain D, and {xn} is a Cauchy sequence in D, is the sequence {f(xn)} also Cauchy ?
This seems to be true. After all, if f is continuous at c, and {xn} is a sequence converging to c, then f(xn) must converge to f(c). And since convergent sequences are Cauchy one would assume that the statement should be true.

But of course life is not so simple. Consider the function f(x) = 1/x and the sequence {1/n}. Then the sequence is convergent to zero, and thus is Cauchy. But f(1/n) = n, which is not Cauchy. This function is continuous in the domain D = (0, 2), say, the sequence {1/n} is Cauchy in D, but the sequence f(1/n) fails to be Cauchy.

The point here is that the function does not need to be continuous at the limit point of a sequence, and hence the above statement is false. While the sequence {1/n} converges to zero, the function f(x) is not continuous at zero. Yet the sequence {1/n} is Cauchy in (0, 2).

Is the statement true if the domain of the function is all of R? Can you find other formulations for which the statement would become true ?

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