You have an interval [a_0, b_0]. You want to find some point x somewhere in this interval. So, you divide the interval into two subintervals, [a_0, c_0] and [c_0, b_0] (usually c_0 = (a_0 + b_0)/2 ) and determine which of these subintervals x lies in. If it’s int he left, you let a_1 = a_0, b_1 = c_0, if it’s in the right you do similarly. You repeat this process, and the a_n and b_n both tend to a common point which is your desired x.

The reason it’s called lion hunting is that the intuitino is you’ve got a lion somewhere in the interval (jungle), and you want to find it. So you stick a net across the jungle and wait till you hear signs of a lion on one side of the net. So now you want to narrow down where the lion is within that part of the jungle…

Anyway, this turns out to be a surprisingly good way to introduce a lot of real analysis, because it basically boils down to doing a binary search for the solution, which is a very intuitive approach to problem solving.

So, want to prove the intermediate value theorem? Do it by lion hunting! Start with f : [a, b] -> R with f(a) = f(a_n) -> f(x) and 0 f(x), so f(x) = 0.

You can also use it to prove bolzano weierstrass. You choose one of the intervals which contains infinitely many terms of the sequence, get x and from the construction there’s a subsequence tending to x.