Definition of convergence of a sequence

Question

Can this be a valid definition?

For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$.

Should I use "$<\epsilon$" or "$\le \epsilon$"?

Answer 1

The following statements

1. For each $\epsilon>0$, there exists $K\in \mathbb{N}$ such that $n\geq K \Rightarrow |a_n-a|\leq \epsilon$

and

2. For each $\epsilon>0$, there exists $K\in \mathbb{N}$ such that $n\geq K \Rightarrow |a_n-a|< \epsilon$

are equivalent.

Proof of 1.$\Rightarrow$2.

For any $\epsilon>0$, we have $\epsilon/2>0$. Therefore by 1., there exists $K\in \mathbb{N}$ such that $n\geq K \Rightarrow |a_n-a|\leq \epsilon/2<\epsilon$, which means 2. is true.

Proof of 2.$\Rightarrow$1.

Trivial because $|a_n-a|< \epsilon$ implies $|a_n-a|\leq \epsilon$