# Prove that $(\forall n\in\Bbb N)(\exists N\in\Bbb N)(\forall m\in\Bbb N)(m\ge\Bbb N \to |(1+\frac{1}{m})^m) - e|<\frac{1}{n})$

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))
• You can learn MathJax from math.meta.stackexchange.com/questions/5020/… – Tengu Oct 29 '17 at 6:13
• what is your definition of $e$ ? – user352653 Oct 29 '17 at 6:14
• It's euler number – Helmi Aziz Oct 29 '17 at 6:15
• 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. – Carl Mummert Oct 30 '17 at 1:41
• So how do I prove that it's less than (1/n)? – Helmi Aziz Oct 30 '17 at 1:56