(Pf) Every Cauchy sequence is bounded

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bwam wrote:

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Recall: a sequence (a_n) is Cauchy iff for all E>0, exists a natural number N such that, for all natural numbers n,m >= N, |a_n - a_m| < E. Prove every Cauchy sequence is bounded.

(Recall: Let (a_n) be a sequence converging to A. We say (a_n) is bounded iff for all E>0, exists a natural number N such that, for all n >= N, |a_n - A| < E.)

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Pf)

Let {a_n} be an unbounded set.
... Then for all k, exist n_k such that |a_n_k| > k.

Then {a_n_k} is an unbounded set.

As such, for all N, exists a_n_k and exists a_n_l such that
... |a_n_k - a_n_l| > |a_n_k| - |a_n_l| = k - l > N.

Thus {a_n} is not Cauchy.
... We have {a_n} unbounded => {a_n} not Cauchy.

The contrapositive gives us {a_n} not Cauchy => {a_n} not bounded.
Hence we conclude every Cauchy sequence is bounded. []

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LostForm wrote:

how about if you have less than 100% avoidance you are bound to get hit. The game you play makes 100% avoidance impossible and EHP has a finite start point. If you play long enough, and the incoming damage is unbounded growth while you play you will eventually die from a hit. A character that 'can not die' in that game relies on boundaries of some sort to incoming damage.

(I don't really understand the proof, except that maybe

Let {a_n} be an unbounded set.
... Then for all k, exist n_k such that |a_n_k| > k

means k isn't > 0)