19.1.2.2 Convergence in metric spaces

Problem: Will the function \( d(x, y) \) \( =\int_{0}^{1} \frac{|x(t)-y(t)|}{e^{t}} d t \) set a metric on \( C([0,1]) \) ? If this function defines a metric, will the sequence \( \left\{x_{n}\right\} \), defined by the formula, \( x_{n}(t)=\left\{\begin{array}{ll}n t, & t \in\left[0 ; \frac{1}{n}\right] \\ 1, & t \in\left[\frac{1}{n} ; 1\right]\end{array}\right. \) be 1) Fundamental; 2) Converging relative to the given metric?