Defined Limit
Example
Prove that,
Proof
We want to show that , such that for all .
Defined Triangle Inequality
Proposition
If the limit of exists, then it is unique!
Prove that,
n→∞lim2n+1n=21.We want to show that ∀ε>0, ∃N>0 such that ∣2n+1n−21∣<ε for all n>N.
2n+1n=2(2n+1)2n−(2n+1)=2(2n+1)1<n1<ε⟺n>ε1.If the limit of xn exists, then it is unique!