Question

Suppose you have n>0 boxes in a row with an inequality sign between each adjoining pair, for example:

Prove, by induction on n, that you can place the numbers 1 through n in these boxes, one per box, in a way that respects the pairwise inequalities. Thus for example, the numbers 1 through 7 could be placed in the above boxes as follows:

Fig: 1

Fig: 2

Fig: 3

Fig: 4