I am trying to prove a statement about an upper bound of a numerical scheme, or recursion, and I need help showing an intermediate result, namely that the sum below is upper bounded by the integral:
$$\Delta t \sum_{\nu = 0}^{n-1}e^{\alpha(t_{n}-t_{\nu+1})} \leq \int_{0}^{t_{n}}e^{\alpha(t_{n}-s)}ds.$$
Now, I have seen that sums of decreasing functions are upper bounded by their corresponding integral version, but how do I go from sum to integral like this? I do not really understand the significance of $\Delta t$ also. Could someone give me a hint/explain how to argue here?
Best regards//
Edit 1 Has there something to do with using right Riemann sums for decreasing functions like this to show the inequality holds?
Edit 2 $\alpha$ is a scalar constant.