Q:

Problem: does sequence of functions converge?

Does the sequence $(f_{n})$, $f_{n} \colon [ -1,0 \rangle \rightarrow \mathbb{R} $,
$$ f_n(x) = 0, x\in [-1,\frac{-1}{n} ],
f_n(x) = nx+1, x\in [\frac{-1}{n},0\rangle $$ converge?
I think yes, $f_{0} = 0, \forall x \in [-1,0\rangle$ with the exception when $x$ is "very close" to $0$, and then $ f_{0} = 1$. As you can see I am having trouble with writing $f_{0}$ down because 0 is not $\in [ -1,0 \rangle $. 

A:

If you fix any point $x \in \left[ -1, 0 \right)$, then we can find a natural number $n_0 \in \mathbb{N}$ such that $x < - \dfrac{1}{n_0}$. This is just the Archemedian property applied in the negative direction. Therefore, after this $n_0$, $f_n \left( x \right) = 0$. Hence, the sequence of functions converges pointwise to the function $f \left( x \right) = 0$ for every $x \in \left[ -1, 0 \right)$. You can check for any analysis book for uniform continuity tests.

