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.