I am studying Bernsteins theorem on completely monotone functions and in the proof I need the uniform convergence of the sequence of functions: for $t>0$ define $h_n(t):=(1-\frac{\lambda t}{n})_{+}^{n-1}$ (where $a_{+}=\max(0,a)$) against the exponentialfunction, i.e. $(t\mapsto e^{-\lambda t})$.
I have tried elementary proofs but cant find a way to proof it. I would be very thankful for help!