R is locally compact since given ϵ>0 and x ∈R, the neighbourhood (x−ϵ,x+ϵ)is contained in [x−ϵ,x+ϵ] and it is known that [x−ϵ,x+ϵ]⊂R is compact. Hence showing that R is locally compact.\text{$\mathbb{R}$ is locally compact since given $\epsilon > 0$ and x $\in \mathbb{R},$ the neighbourhood $(x-\epsilon, x+\epsilon )$}\\ \text{is contained in $[x-\epsilon, x+\epsilon ]$ and it is known that $[x-\epsilon, x+\epsilon ] \subset \mathbb{R}$ is compact. }\\ \text{Hence showing that $\mathbb{R}$ is locally compact.}R is locally compact since given ϵ>0 and x ∈R, the neighbourhood (x−ϵ,x+ϵ)is contained in [x−ϵ,x+ϵ] and it is known that [x−ϵ,x+ϵ]⊂R is compact. Hence showing that R is locally compact.
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