I want to determine the value of the following term
$$\inf \bigg\{\sup \bigg\{ \frac{2n}{2+n(-1)^n}:n>k\bigg\}:k\in\mathbb N\bigg\}$$
The main problem why I cannot solve this by my own is that the $(-1)^n$ is really disturbing me. Its hard for me to control the bevaviour of the inner term (hard to now what the supremum is).
If anyone could help me, I would be really happy.
On
Recall the following first (it is for some people the definition of $\limsup$) : \begin{align} \limsup_{n\to\infty} a_n = \inf_{k\in\mathbb{N}} \left(\sup_{n>k} a_n\right) \end{align} Then it is easy: \begin{align} \limsup_{n\to\infty} a_n = \limsup_{n\to\infty} \frac{2n}{2+n(-1)^n} = 2 \end{align} Conclude.
For a fixed $k\in \mathbb{N}$, the limit of $\frac{2n}{2+n(-1)^n} | n>k$ as $n\to \infty$ is $(-1)^n 2$, hence the supremum is clearly $2$, and note that this is independent of the value $k$.
Therefore, the inner set generates always the value $2$ independent of the value of $k$, hence the infinitum is again 2.