One way to think about limits
Citat (ungefär): We say "a_n converges to l" if for all epsilon > 0, there exists N such that for all n > N, |a_n - l| < epsilon.
Think of it as a game between you and an adversary. No matter how he chooses the goalposts around l, you must be able to give an N such that all succeeding terms stay within those goalposts. If your enemy can find two goalposts such that infinitely many terms lie outside them, THEN YOU'VE LOST THE GAME.
Think of it as a game between you and an adversary. No matter how he chooses the goalposts around l, you must be able to give an N such that all succeeding terms stay within those goalposts. If your enemy can find two goalposts such that infinitely many terms lie outside them, THEN YOU'VE LOST THE GAME.
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