How to judge the convergence of abstract sequence

Given that ${u_n}$ is a monotonically increasing bounded sequence, now I want to judge whether the following sequence converges:

$$begin{array}{c}
A. sum_{n = 1}^{infty} frac{u_{n}}{n} & B. sum_{n = 1}^{infty}(-1)^{n} frac{1}{u_{n}}
C. sum_{n = 1}^{infty}left(1-frac{u_{n}}{u_{n+1}}right) & D. sum_{n = 1}^{infty}left(u_{n+1}^{2}-u_{n}^{2}right)
end{array}$$

Since ${u_n}$ is an abstract sequence, I don’t know how to use MMA to solve this problem.

Therefore, I use the following special case to verify the problem:

u[n_] := Sqrt[n/(n + 1)]

Sum[u[n]/n, {n, 1, Infinity}]
Sum[(-1)^n/u[n], {n, 1, Infinity}]
Sum[1 - u[n]/u[n + 1], {n, 1, Infinity}]
Sum[u[n + 1]^2 - u[n]^2, {n, 1, Infinity}]

However, MMA cannot judge whether Sum[1 - u[n]/u[n + 1], {n, 1, Infinity}] converges or not.

How can I judge the convergence of option C correctly?

Clear["Global`*"]

u[n_] := Sqrt[n/(n + 1)]

SumConvergence[1 - u[n]/u[n + 1], n]

(* True *)

Assuming[n > 0, {#, SumConvergence[
      1 - u[n]/u[n + 1] // Simplify, n, Method -> #]} & /@
   {"RatioTest", 
    "RootTest", "RaabeTest", "IntegralTest"}] //
 Grid[#, Frame -> All] &