Final answer:
A Cauchy sequence in C1[0,1] is a sequence of continuously differentiable functions on the interval [0,1] that become arbitrarily close to each other. C1[0,1] is a complete space, meaning every Cauchy sequence should converge to an element within this space. If the sequence converges to a function not continuously differentiable everywhere on [0,1], it would not belong to C1[0,1], but this would be a contradiction since all Cauchy sequences in C1[0,1] should converge within the space.
Step-by-step explanation:
In analysis, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Specifically, in the space of continuous functions C1[0,1], which contains all continuously differentiable functions on the interval [0,1], a sequence (fn) of functions is considered a Cauchy sequence if for every ε > 0, there exists an N such that for all m, n > N, the maximum difference between fm and fn on [0,1] is less than ε. Now, in a complete metric space, every Cauchy sequence would converge to an element within the space. However, since the question asks for a Cauchy sequence in C1[0,1] that does not converge, this seems to be a trick question as C1[0,1] is a complete subset of function spaces, meaning every Cauchy sequence should converge within this space. But, it may be possible that the sequence converges to a function that is not differentiable everywhere on [0,1], making the limit outside of C1[0,1]. Thus, if the asking is towards constructing such a sequence, it is a contradiction in terms; every Cauchy sequence in C1[0,1] should converge to a continuously differentiable function on [0,1], which belongs to C1[0,1]. Without further specific constraints or context, typically, all Cauchy sequences in C1[0,1] should indeed converge within the space.