Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

No, the continuous image of a Cauchy sequence is not necessarily a Cauchy sequence. To understand this, we first recall that a Cauchy sequence (xn)(x_n) in a metric space XX is one where, for every ϵ>0\epsilon > 0, there exists an N∈NN \in \mathbb{N} ___ ____ _______ ______ ________ __________ ______ _________ _______ ________ ______.
______ ____ _______ ___ _______ ______ ___ ___ ______ ______.
_______ ___ ________ ______ _____ _____ ________ ___ ____ ___ ____.
_________ __________ ____ _______ __________.
______ _____ ____ ______ ___ __________ __________ _____ _________ ___ __________.
_________ ________ _______ _________ __________ ___.
________ ___ ______ _______ __________.
_____ ________ __________ _________ ______ _____ _____ _____.
________ _____ ___ __________ ________ _____.
_______ ____ _____ ___ ____ _______ ________ ____.
__________ ______ ________ _____ _______ ______ _______ ___.
_____ _________ ________ _____ _____ _____ ________.
________ ___ _________ ______ __________ ____ ________ ____.
___ ___ _______ _______ _____ _________ _______ _______ ___ _____ __________.
________ _________ ________ _____ ___ _______ _______ __________ ______ ___.
____ ___ _______ ______ __________ __________ ___.
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