Examples of u_n and v_n Sequences with Finite Sum | Help with Homework"

[mercedesbenz]

Homework Statement

Please help me find examples.

u_n\subset [0,\infty) and v_n\subset [0,\infty) such that
u_{n+1}\leq u_n+v_n for all n and \sum_{n=1}^{\infty}v_n
is finite.

Homework Equations

The Attempt at a Solution

 

[morphism]

Say you take u_n=0 for all n...

 

[mercedesbenz]

Say you take u_n=0 for all n...

I want to find v_n\neq 0 ,u_n\neq 0 for all n

 

[morphism]

My advice is to first pick a nice, summable v_n.

 

[mercedesbenz]

Please help me find examples.

u_n\subset [0,\infty) and v_n\subset [0,\infty) such that
u_{n+1}\leq u_n+v_n for all n and \sum_{n=1}^{\infty}v_n
is finite.

In fact, this is a theorem which say that

u_n\subset [0,\infty) and v_n\subset [0,\infty) such that
u_{n+1}\leq u_n+v_n for all n
If \sum_{n=1}^{\infty}v_n
is finite then \displaystyle{\lim_{n\rightarrow \infty}u_n} exists

then I want to find u_n which dificult to find lim in order to guarantee
this therem is well better than MCT because this therem is generalization of MCT.

 

[morphism]

Why don't you want to put in some effort?

Pick a summable (v_n), like say v_n = 1/2^n. Now pick any decreasing (u_n), like u_n = 1/n.