Keeping Up The Pressure

Keeping Up The Pressure

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Hi chess friends,

I've posted a few times recently about defending difficult positions. To ballance things out I want to show an example of an endgame where a small advantage can slowly grow and turn into a big one and finally a win. While it can take a long time to convert a pawn or a positional advantage, it helps to keep in mind that it's probably more unplasant for the opponent who may have to take hours and can only hope for a draw.

In the following game I emerged from the opening ahead by the pawn that my opponent, an experienced master, had sacrificed. I had king safety issues and was far from promoting the pawn, but I was slowly able to unwind, keep the extra pawn and promote. I'll first include a couple of warm-up puzzles and then the game itself. 

This was white's final chance to completely equalize and force a draw. What would you play?

In the position below white is threatening to win the a-pawn. How can black keep the material advantage?
Here's the full endgame. The analysis covers some theoretical rook endgames, where I'm not 100% sure about my conclusions. Any improvements to the analysis that you can find are appreciated!
Enjoy.