I am trying to find a smooth and differentiable function, which converges to the characteristic function $\mathbf{1}_{[a,b]}$. Is there an easy idea how to realise that? I suppose it should look like this $$ f_n(x)= \left\{ \begin{array}{ll} 1 & x \in [a,b] \\ 1-n(x-b) & \, x\in [b,(1/n)+b] \\ n(...

In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can...

art, really? What is it good for? Do we really need it?