Prove a functional series is pointwise and/or uniformly convergent

Question

Determine whether the following sequence is pointwise and/or uniformly convergent

$(f_n)_{n\in\mathbb{N}}$ where $x\in\mathbb{R}$ and

$$f_n(x)= \left\{ \begin{array}{ll} n & x\geq n \\ 1 & x< n \\ \end{array} \right. $$

So I have concluded that $f_n(x)\rightarrow 1$ as $n\rightarrow\infty$ so the limit function is $f(x)=1$

How would i know prove or disprove uniform convergence? The way the function is shown with $x\leq n$ etc confuses me

Answer 1

With $f(x):=1$ $$||f_n- f||_\infty = n-1$$

so no uniform convergence.

However $$||f_n- f||_{\infty,[0,R]} = 0 $$

for $n>R$, so uniform convergence on bounded intervals.