Re: Question about limits

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Re: Question about limits

Quote:

Originally Posted by

**MagisterMan** |bn - L| < ε => L=M

L = M does not follow from the fact that |bn - L| < ε for *some particular* n and ε.

The main problem with your proof is that it silently goes from statements that quantify over n and ε to statements about some particular n and ε and then back. If you say "For every ε there exists N_{1} such that n >= N1 implies L - ε < an < L + ε," the variables ε and n exist only inside that phrase. Once you finish the sentence, you can't refer to ε or n. Therefore, you can't continue by writing "L - ε < an < L + ε."

Every variable in a proof must be properly introduced. If you are proving a statement of the form "For all n, ...," you can say "Fix n" and use that n for the rest of the proof of that statement. If you have an assumption or a proven fact of the form "There exists an n such that ...," you can say "Consider an n such that ..." and again use it until the end. A third way to introduce a variable is by an explicit definition: e.g., "Let ε = M - L." Then you can instantiate an assumption or a proven fact of the form "For every ε, ..." using this particular ε. In this particular problem, choosing a proper ε is an important step.

Re: Question about limits

Thanks **emakarov** for the clarification