Real numbers/Subsets/Function/Limit/Characterizations/Fact/Proof/Exercise

Let ${}T\subseteq \mathbb {R}$ be a subset and let ${}a\in \mathbb {R}$ be a point. Let ${}f\colon T\rightarrow \mathbb {R}$ be a function and ${}b\in \mathbb {R}$. Prove that the following statements are equivalent.

1. We have

   ${}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$

2. For all ${}\epsilon >0$ there exists a ${}\delta >0$ such that for all ${}x\in T$ with ${}d(x,a)\leq \delta$ the inequality ${}d(f(x),b)\leq \epsilon$ holds.

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