So I got this problem from my professor. For the first question, I've tried to wrote m as n or (n+1) (because m was supposedly greater than or equal to N, which was a part of n. Correct me if I'm wrong) , but still can't find a solution. And for the second question, I don't even know where to start.

(I'm sorry I can't write down the question properly with mathjax)

  1. (∀n∈ℕ)(∃N∈ℕ)(∀m∈ℕ)(m≥N → |(1+(1/m))^m) - e|<(1/n))
  2. (∀x∈ℚ)(∃n∈ℕ)(∀N∈ℕ)(∃m∈ℕ)(m≥N & |(1+(1/m))^m - x|≥(1/n))
  • $\begingroup$ You can learn MathJax from math.meta.stackexchange.com/questions/5020/… $\endgroup$ – Tengu Oct 29 '17 at 6:13
  • $\begingroup$ what is your definition of $e$ ? $\endgroup$ – user352653 Oct 29 '17 at 6:14
  • $\begingroup$ It's euler number $\endgroup$ – Helmi Aziz Oct 29 '17 at 6:15
  • $\begingroup$ The first part of the question from the professor just wants you to prove that the sequence $$1 + (1/m))^m$ converges to $e$. It is a problem from elementary calculus, not from mathematical logic. The proof may not be obvious, depending on your definition of $e$, but it is covered in many texts. $\endgroup$ – Carl Mummert Oct 30 '17 at 1:41
  • $\begingroup$ So how do I prove that it's less than (1/n)? $\endgroup$ – Helmi Aziz Oct 30 '17 at 1:56

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