Increasing function/R to R/Sequence limit and function limit/Exercise

Let

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

be an increasing function and ${}b\in \mathbb {R}$. Show that the sequence ${}f(n)$, ${}n\in \mathbb {N}$, converges to ${}b$ if and only if

${}\operatorname {lim} _{x\rightarrow +\infty }\,f(x)=b\,$

holds, i.e. if the limit of the function for ${}x\rightarrow +\infty$ is ${}b$.

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