Prove the following statements: a. There is a natural number M so that for all r

Question

Prove the following statements:

a. There is a natural number M so that for all real numbers r > M, (1/r) < 0.13

b. For every pair of positive real numbers x and y where x < y, there exists a natural number M so that for all real numbers r > M, (1/r) < (y - x)

This is for an introductory proof class, so the proof should not be overly complicated, or longer than a short paragraph. I'm stuck on both of them, so a clear explanation will help a lot! Thanks!!!

Explanation / Answer

If r > M and they are both positive, then 1/r < 1/M. So, all we have to do is make 1/M 0.13. That is, have M 100/13. So, we can pick a natural number for M as long as it is larger than 100/13. Proof: Let M = 10. If r> M, then 1/r < 1/M = 0.1 < 0.13, since both r and M are positive.

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