1. Feb 4, 2008

### mercedesbenz

1. The problem statement, all variables and given/known data

$$u_n\subset [0,\infty)$$ 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.

2. Relevant equations

3. The attempt at a solution

2. Feb 4, 2008

### morphism

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

3. Feb 4, 2008

### mercedesbenz

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

4. Feb 5, 2008

### morphism

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

5. Feb 5, 2008

### mercedesbenz

$$u_n\subset [0,\infty)$$ 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.

6. Feb 5, 2008

### 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.