Quiz, week 3

Calculate the limit of $a_n = d_∞(f_n, f_{n+1})$, where $f_n(x) = x/n$ and $f_n : [0, 1] → \mathbb{R}$.

Consider the metric space $(\mathbb{R}_{++}, d)$ where $d(x, y) = |1/x - 1/y|. Is $x_n = 1/n$ a Cauchy sequence in this space? What about $x_n = n$?

Consider the following premise: $f$ is a weakly monotone function. Which combination of cases is exhaustive:

1. $f$ is strictly increasing.
2. $f$ is weakly increasing.
3. $f$ is weakly decreasing.