Finding sequences u_n and v_n where u_{n+1}≤u_n+v_n and Σv_n converges

[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.