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1) why must you find a [itex]n_0 \in N[/itex] such that [itex]\forall N \geq n_0[/itex]? You might as well say find a [itex]n_0 \in R[/itex] such that [itex]\forall N > n_0[/itex]. Just a matter of simplicity?

2) Why must [itex]|x_n - a| < \epsilon[/itex] hold? I think [itex]|x_n - a| \leq \epsilon[/itex] is fine as well, given that it must hold [itex]\forall \epsilon > 0[/itex].