Card guessing game

Hint: The last suit to get its first card picked will necessarily be guessed correctly.

Think of it this way:

Let's say there's a situation when there are still $13$ hearts in the deck, but there are fewer than $13$ cards of the other three suits (such a situation will surely arrive - you can prove it). But that means the player will guess "Heart" every time until the heart is actually drawn, so he will certainly guess at least one correct card, and there will, after that, still be $12$ hearts in the deck.

Now, you are left with $12$ hearts in the deck, and an unknown number of other suits. If there are fewer than $12$ spades, clubs and diamonds, the same argument from above applies. If not, keep drawing cards until only one suit has $12$ representatives in the deck.

Can you see the pattern?

Show that before and after each card is drawn, the sum of number of correct guesses already made and the length of the longest suit remaining in the deck is always at least $13.$