A positive integer $N\ge 2$ is given. A collection of $N(N+1)$ soccer players, no two of the same height, stand in a row. Sir Alex wants to remove $N(N-1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: no one stands between the two tallest players, no one stands between the third and fourth tallest players, no one stands between the fifth and sixth tallest players, $\dots$, no one stands between the two shortest players. Show that this is always possible.

Step 1 (Reformulation): Label players $1,2,\dots ,N(N+1)$ by height (tallest = 1). We need to select $2N$ players from the row such that players ranked $\{1,2\}$, $\{3,4\}$, $\dots$, $\{2N-1,2N\}$ are adjacent in the selected subsequence (no selected player between them).

Step 2 (Key lemma): Among any $k+1$ consecutive players (by rank) in a row of $m$ players, we can find two that are adjacent in the row (no other of these $k+1$ players between them) as long as $m$ is large enough.

Step 3 (Greedy approach): We pair off the players by rank: $(1,2),(3,4),\dots ,(2N-1,2N)$. We select these pairs greedily. Among the $N(N+1)$ players, consider the positions of players ranked $1$ and $2$. They divide the row into at most 3 segments. We select them as our first pair.

Now we need to select the remaining $N-1$ pairs from the players between/around the first pair, maintaining the adjacency condition. The key insight is that with $N(N+1)$ players and only $2N$ to select, we have $N(N-1)$ players to remove, providing enough room to find valid adjacent pairs at each step.

Step 4 (Formal induction): By induction on $N$. The base case $N=2$ requires selecting 4 players from 6 such that the top-2 and bottom-2 are each adjacent. This can be verified directly. For the inductive step, find the pair $(2N-1,2N)$ (the shortest pair) — among the $N(N+1)$ players, the positions of these two split the row. Select them, then apply the inductive hypothesis to the remaining configuration with $N-1$ pairs and $(N-1)N$ remaining relevant players.