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Cutting the square

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I don't know how simple/diffcult this is, but it has been on my mind for some time, and I haven't managed to do anything about it, so now I'll try here to see if anyone has any input.

Let a natural number $n\in\mathbb N$ be given.

Start with a square $S_0$ with sidelength $1$.

For $k$ in $\{1,2,\ldots, n-1\}$, define $B_k$ as a section of $S_{k-1}$ of area $\frac{1}{n}$ defined by a straight line, and $S_k$ as $S\setminus B_k$. (I believe a simple continuity argument proves the existence of $B_k$.)

And set $B_n = S_{n-1}$.

(If anything in the process is unclear, it might help to know it was inspired by cutting a slice of bread)

What is $\inf\left\{\sup\left\{\operatorname{diam}(B_k)\mid k\in \left\{1,2,\ldots,n\right\}\right\}\mid\text{every set of }B_k\text{'s}\right\}$?

(I don't find it easy to parametrise the set of $B_k$'s, but I hope it's understandable.)

(Where $\operatorname{diam}$ is the diameter of a set, defined as $\operatorname{diam}(A) = \sup\{d(x,y)\mid x,y\in A\}$ - I think that's standard in metric spaces)

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