Let (i,j) be an open interval. Let a∈(i,j) . We show that, we can find ϵ>0∋ N(a,ϵ)⊂(i,j)
Since a∈(i,j) we have that i<a<j this implies that a−i>0 and j−a>0
Set ϵ=min(a−i,j−a)
Suppose further that b∈N(a,ϵ) this implies that ∣b−a∣<ϵ
−ϵ<b−a<ϵa−ϵ<b<a+ϵ
And since ϵ is as defined we have that a−ϵ≥ia+ϵ≤j
This implies that b∈(i,j)
Hence N(a,ϵ)⊂(i,j)
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