Pointwise convergence can lose boundedness and need not preserve sup norms.Suggest a slogan
For \(n\geq1\), the functions \(f_n\colon(0,1)\to\RR\), \(f_n(x)\coloneqq\min(n,1/x)\), are bounded and converge pointwise to the unbounded function \(x\mapsto1/x\).
For \(n\geq1\), the functions \(g_n\coloneqq\chi_{[n,n+1]}\colon\RR\to\RR\) converge pointwise to \(0\), while \[ \lim_{n\to\infty}\sup_{x\in\RR}\abs{g_n(x)}=1\neq0=\sup_{x\in\RR}\abs{\lim_{n\to\infty}g_n(x)} . \]