I am new to proofs and would appreciate advice on this proof.
Prove that $\lim_{x\to-\infty} \frac1x = 0$.
Given $\epsilon>0$ find N such that:
if $x < N$ then $\left\lvert\frac1x – 0\right\rvert<\epsilon$
choose $N = -{1\over \epsilon}$.
Since ${x\to -\infty}$, $x < -{1 \over \epsilon}$, then $\left\lvert \frac1x – 0 \right\rvert = \left\lvert \frac1x \right\rvert = {1 \over -x} < \epsilon \Rightarrow x < -{1 \over \epsilon}$ qed.
Is this proof correct, and if not, where did I go wrong? Will also appreciate advice on the construction of the proof. Thank you.
Best Answer
You need to reorder some things in your proof so that the wording is logical.
For example, instead of "Since $x \rightarrow -\infty$ ... ," something along the following lines would be more clear: