Real sequence/Unit fractions/Continuous and convergent/Function limit/Exercise

Let

${}T={\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\subseteq \mathbb {R} \,$

be the set of the unit fractions and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a real sequence. Let ${}b\in \mathbb {R}$ and ${}D=T\cup \{0\}$. Show that the following properties are equivalent.

1. The sequence converges to ${}b$.
2. The function

   $f\colon T\longrightarrow \mathbb {R}$

   given by

   ${}f{\left({\frac {1}{n}}\right)}=x_{n}\,$

   has a limit ${}\operatorname {lim} _{x\rightarrow 0}\,f(x)=b$.

3. The function

   ${\tilde {f}}\colon D\longrightarrow \mathbb {R}$

   given by

   ${}{\tilde {f}}{\left({\frac {1}{n}}\right)}=x_{n}\,$

   and ${}{\tilde {f}}(0)=b$ is continuous.

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