Formally, given a metric space $(X, d)$, a $x_1, x_2, x_3, ...$ is Cauchy, if for every positive real number $ε > 0$ there is a positive integer $N$ such that for all positive integers $m, n > N$, the distance, $d(x_m, x_n) < ε$.
- suppose $f_n:(X,d_x)→(Y,d_y)$ for all $n$.
- If a sequence of functions is a cauchy sequence in $(Y,d_y)$, what does this mean? Does the positive number $N$ depends on $x$?
Assuming that you are working in the space of bounded and continuous functions from $X$ to $Y$ and that the distance in that space is$$d(f,g)=\sup\left\{d_Y\bigl(f(x),g(x)\bigr)\,\middle|\,x\in X\right\},$$then a sequence $(f_n)_{n\in\mathbb N}$ is a Cauchy sequence if and only if$$(\forall\varepsilon>0)(\exists p\in\mathbb{N})(\forall m,n\in\mathbb{N})(\forall x\in X):d_Y\bigl(f_m(x),f_n(x)\bigr)<\varepsilon.$$