If $(a_n)$ is a decreasing sequence of positive real numbers and $(u_n)$ is a strictly increasing sequence
of positive integers such that $ \dfrac {u_{n+2}-u_{n+1}}{u_{n+1}-u_n}$ is bounded , then how do we prove that the
convergence of $\sum_{n=1}^ \infty (u_{n+1}-u_n)a_{u_n}$ implies the convergence of $\sum_{n=1}^ \infty a_n$ ?
Suppose that $$ \frac{u_{n+2}-u_{n+1}}{u_{n+1}-u_n}\le M\tag{1} $$ Simply counting terms and using the monotonicity of $a_k$, we get $$ (u_{n+1}-u_n)a_{u_{n+1}}\le\sum_{u_n\le k\lt u_{n+1}}a_k\le(u_{n+1}-u_n)a_{u_n}\tag{2} $$ summing $(2)$ yields $$ \sum_{n=1}^\infty(u_{n}-u_{n-1})a_{u_n}\le\sum_{k=1}^\infty a_k\le\sum_{n=0}^\infty(u_{n+1}-u_n)a_{u_n}\tag{3} $$ Applying $(1)$ gives $$ \frac1M\sum_{n=1}^\infty(u_{n+1}-u_{n})a_{u_n}\le\sum_{k=1}^\infty a_k\le\sum_{n=0}^\infty(u_{n+1}-u_n)a_{u_n}\tag{4} $$
