Understanding limsup

Solution 1:

It means "weakly" decreasing, so they may not get smaller, but they don't get larger. Let's look at $3-1/n$. \begin{align} & \sup\left\{3 - \frac1 1, 3-\frac12,3-\frac13,3-\frac14,3-\frac15,\ldots\right\} \\[8pt] & \sup\left\{ 3-\frac12,3-\frac13,3-\frac14,3-\frac15,\ldots\right\} \\[8pt] & \sup\left\{3-\frac13,3-\frac14,3-\frac15,\ldots\right\} \\[8pt] & \sup\left\{3-\frac14,3-\frac15,\ldots\right\} \\[8pt] & {}\qquad \vdots \end{align}

The sequence of suprema gets (weakly) smaller.

Solution 2:

When I first learned about $\limsup$ and $\liminf$ I found it helpful to think of them as, respectively, "eventual least upper bound" and "eventual greatest lower bound". They are variants of the LUB and GLB that avoid taking account of any finite number of (possibly "deviant") terms of the sequence, but only record the limiting trend in how high or how low the terms get.

If you take any set $S$ of real numbers and remove some values to get a smaller set $S' \subset S$, then $\sup S' \leq \sup S$ by definition (any upper bound of $S$ is an upper bound of $S'$, in particular the LUB of $S$). The key point, which I think you are missing, is that in the definition of $\limsup a_n$, the $s_N$ are defined to be over all $n \geq N$, so for an increasing sequence like the one you gave, all the $s_N$ are equal (in this case to $3$).

Solution 3:

I know you already accepted an answer, but nothing does it like a picture for me. Imagine our sequence does something like this where $n$ increases in the positive $x$ direction. a fluctuating sequence.