I have a basic analysis problem that I don't manage to solve:
Let $(a_n)_{n \in \mathbb{N}}\subseteq \mathbb{R}$ be a strictly positive sequence with $a_n \overset{n \to \infty}{\longrightarrow}\infty$ and $(u_n)_{n \in \mathbb{N}}\subseteq \mathbb{R}$ be a sequence converging to $u\in \mathbb{R}$.
I wish to show that $$\lim_{n \to \infty}\frac{1}{a_n}\sum_{k=1}^n (a_k-a_{k-1})u_k=u$$
What I observe is that $\frac{1}{a_n}\sum_{k=1}^n (a_k-a_{k-1})=1$. I tried to apply Hölder's inequality but got stuck.
Could anyone give me a hint how to approach the problem?
Thank you so much!
As you pointed out,
\begin{equation} \frac{1}{a_n}\sum_{k=1}^{n}(a_k-a_{k-1})=1, \end{equation}
thus
\begin{equation} \frac{1}{a_n}\sum_{k=1}^{n}(a_k-a_{k-1})u=u. \end{equation}
Now, let $\epsilon >0$ be arbitrary. There exists $N_\epsilon\in\mathbb{N}$, such that $\forall n>N_\epsilon$ we have $|u_n-u|<\epsilon$. We want to estimate
\begin{equation} \frac{1}{a_n}\sum_{k=1}^{n}(a_k-a_{k-1})(u_k-u), \end{equation}
so let $M\in\mathbb{N}$ be large, in particular $M>N_\epsilon$. Then,
\begin{align} |\frac{1}{a_M}\sum_{k=1}^{M}(a_k-a_{k-1})(u_k-u)|&< |\frac{1}{a_M}\sum_{k=1}^{N_\epsilon}(a_k-a_{k-1})(u_k-u)|+|\frac{1}{a_M}\sum_{k=N_\epsilon +1}^{M}(a_k-a_{k-1})(u_k-u)| \\ &<\frac{C}{a_M}+\epsilon |\frac{1}{a_M}\sum_{k=N_\epsilon +1}^{M}(a_k-a_{k-1})| \\ &<\frac{C}{a_M}+\frac{\epsilon}{a_M}|a_M-a_{N_\epsilon}| \\ &<\epsilon+\frac{C+\epsilon a_{N_\epsilon}}{a_M} \overset{M\rightarrow +\infty}{\longrightarrow} \epsilon \end{align}
Edit: This is not exactly precise, but the idea should work. Edit 2: Typos