Add punctuation: The first seven conditions can be shown to be satisfied for the equivalence classes of convergent sequences as a result of the rational numbers satisfying those conditions
A: The first seven conditions can be shown to be satisfied for the equivalence classes of convergent sequences as a result of the rational numbers satisfying those conditions.

Add punctuation: We were so fortunate to be able to visit them
A: We were so fortunate to be able to visit them.

Add punctuation: Photos Peterhof Palace Russia photos of Russia Russia pictures
A:
Photos - Peterhof Palace , Russia, photos of Russia, Russia pictures