# Point wise convergence of Laplace transform and convergence of functions

Assume that functions $$f_n(t), f(t)\in C_b(R_+)$$. For every $$\lambda >0$$, we have $$\bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1},$$ where $$C_\lambda>0$$ is a constant. Can we get that for every $$t>0$$, $$f_n(t)$$ converges to $$f(t)$$? Is it possible to get the order of convergence?

• If $f_n(t) = \sin(n t)$ and $f(t) = 0$, then the left-hand side is $n/(\lambda^2 + n^2)$, which is $O(1/n)$ despite the fact that $f_n(t)$ does not converge to $f(t)$ pointwise. Or am I missing something? – Mateusz Kwaśnicki Feb 25 '20 at 21:38