No.
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No, but if you have one you can plug a PS1 memory card into a PS2 and save it on that
In order to play PS1 games on a PS2 you need to have a PS1 memory card in your PS2, so you can save your progress. Otherwise they don't work.
I do believe that a PlayStation 2 will recognize a PlayStation 1 memory card. I use a ps1 memory card in my ps2 might it might vary from the different memory card brands.
i'd doubt it, a ps2 is bigger than a ps1, therefore the card may not even fit.
You must have a PS1 memory card.