Balanced Cards 1000 pts

There are $n\ge 1$ cards, each with a red number and a blue number. The red numbers are $1,2,\ldots,n$, each used exactly once. The blue numbers are $1,2,\ldots,n$, each used exactly once.

Prove that the cards can be picked up one at a time, each exactly once, so that after every pick the red and blue sums on the cards picked up so far differ by at most $n-1$.