Telescoping Series: If only $\liminf$ does exist, and not $\lim$

Question

Consider the following telescoping series:

$$S:=\sum_{t=0}^{\infty} (x_t - x_{t+1}).$$

If $$\lim_{t \rightarrow \infty} x_t=0,$$ then this simplifies to

$$S=x_0 - \lim_{t \rightarrow \infty} x_t=x_0.$$

However, if the limit does not exist, but under the assumption that $$\liminf_{t \rightarrow \infty} x_t = 0,$$ we want to show that

$$S \geq x_0.$$

Can you help me with this? (I have not studied maths). Thank you very much!

Answer 1

You cannot prove it, because it is not true. Consider $x_t=0$ if $t$ is odd and $x_t = 1$ if $t$ is even. Then $$\sum_{t=0}^{\infty} (x_t - x_{t+1})$$ does not converge, because $$\lim_{n \to \infty}\sum_{t=0}^{n} (x_t - x_{t+1})$$ does not exist.