I have the function sequence: $$f_{n}(x)=\chi_{(n,\infty)}(x)=\begin{cases}1, & \text{ if } x>n\\0, & \text{ otherwise}\end{cases}$$ Why does this sequence converges almost everywhere on $\mathbb R$ to $0$? So the measure of the set when $f_{n}$ does not converge to $0$ shouldn't be infinite? And I don't understand why this sequence does not converge almost uniformly to $0$.
Please help me with this!
Define $a_n=\sup_x{|f_n(x)-0|}$ as here. The sequence $f_n$ converges uniformly iff $a_n \to 0$. But for each $n$ there exists $x_n>n$ and therefore $f_n(x_n)=1$ which implies $a_n=1$ for each $n\mathbb \in N$. Hence $$\lim_{n\to \infty} a_n=1$$ and therefore the sequence does not converge almost uniformly to $0$. Contrary the sequence converges pointwise to $0$. For each $x \in \mathbb R$ there exists $n_x \in \mathbb N$ such that $x<n_x$. Then it is the fact that $$\lim_{n\to +\infty}f_n(x)=\lim_{n\to+\infty}f_{n_x+n}(x)=0$$