Suppose $X$ and $Y$ be metric spaces.Let $f:X\to Y$ be function which is Cauchy Continuous.Show that $f$ is continuous.
Cauchy Continuity:Let $X$ and $Y$ be metric spaces, and let $f$ be a function from $X$ to $Y$. Then $f$ is Cauchy-continuous if and only if, given any Cauchy sequence $<x_n>$ in X, the sequence $<f(x_n)>$ is a Cauchy sequence in Y.
I am planning to use sequential definition of Continuity but I am unable to produce a neat proof without completeness.Give me some idea!
