## Sunday, February 19, 2012

### Confusing lim sup and lim inf

The definitions of $\limsup$ and $\liminf$ can get super confusing. I h0pe typing it out will help me remember it once and for all.

Let $\{x_n\}$ be a sequence of real numbers indexed by $n \in \mathbf{N}$. Consider the sequence $y_k = \sup_{n \geq k} x_n$ which is formed using the supremums of tails of the original sequence. Observe that $y_k$ is non-increasing (since the supremum over a smaller set can only get smaller''). We define$\limsup x_n := \inf_k y_k = \inf_k \sup_{n\geq k} x_n$.

Now if we switch the order of infimum and supremum in the above definition, the new quantity still makes sense (because the sequence of infimums of tails is non-decreasing). It is natural to define $\liminf x_n := \sup_k \inf_{n\geq k} x_n$.

When faced with $\limsup$, think of a sequence of supremums obtained by sequentially chopping off the initial terms of the given sequence. Then take its limit, i.e., its infimum. One can interpret $\liminf$ in a similar way.