Question related to the proof of "If $c>0$, then $\lim {c^{1/n}}=1$"

Question

To do this proof, for $c>1$, $c^{1/n}=1+d_n$, and for $0<c<1$, $c^{1/n}=\frac{1}{1+h_n}$, where $d_n>0$, and, $h_n>0$ respectively.

I am having trouble understanding why we can write these equations. Which rules assert these?

Without knowing the rules, these equations seem to appear out of blue.

Answer 1

It is because if $c>1$, then $c^{1/n}>1$, so we let $d_{n}=c^{1/n}-1$, then it turns out that $c^{1/n}=1+d_{n}$. Similarly, since $0<c<1$, then $0<c^{1/n}<1$, then $\dfrac{1}{c^{1/n}}>1$ and we let $h_{n}=\dfrac{1}{c^{1/n}}-1>0$.