Section 8

Examples of formal proofs:

Ex 1

A proof structure for these types of proofs is something like

1. Assume $ϵ>0$
2. Specify some $N$ and verify the limit convergence theorem.
3. Then $n>N⟹|s_n-s|<ϵ$

For these proofs you will mostly have to work backwards from your conclusion, then write out the proof. Lets see both the scratch work and then the formal proof for the following:

Prove $\lim \frac{1}{n^2}\to 0$

We want to show that $|\frac{1}{n^2}-0|<ϵ$ This is the same as saying $\frac{1}{n^2}<ϵ$ Then $\frac{1}{ϵ}<n^2$ $\frac{1}{\sqrt{ϵ}}<n$ Let $N=\frac{1}{\sqrt{ϵ}}$. Then we know that $n>N⟹|\frac{1}{n^2}-0|<ϵ$

Lets write this as a formal proof: Proof:

1. Assume $ϵ>0$
2. Then, lets pick some $N=\frac{1}{\sqrt{ϵ}}$
3. Then $n>N⟹n>\frac{1}{\sqrt{ϵ}}$
4. This means, $n^2>\frac{1}{ϵ}⟹ϵ>\frac{1}{n^2}$
5. So from here we can prove, $\frac{1}{n^2}<ϵ⟹|\frac{1}{n^2}-0|<ϵ$
6. Thus, for $n<N=\frac{1}{\sqrt{ϵ}}$ we know $|\frac{1}{n^2}-0|<ϵ$

todo there are a couple more here I didnt understand, along side the hw. Make sure to finish and understand them.