Banach's Contraction Mapping Principle theorem

Question

can you please some one explain me how the highlighted statement comes

im understanding all but not the last one please

Answer 1

From the inequality above the highlighted one, for every $1\leq n < m$, one can see that $$\sum_{i=n}^md(T^i,T^{i+1}) \leq \phi(T^n(x))-\phi(T^{m+1})\leq \phi(T^n(x)).$$ Where the last inequality holds because $\phi$ takes values in $\Bbb R^+=[0,\infty)$. Letting $m\to \infty$ gives you the desired expression.

Answer 2

Since $d$ is a metric, and $k$ is in $(0,1)$, we can say that $\varphi$ is a non-negative function. Hence what you have by the inequality before is $\sum_{i=1}^{\infty}d(T^i(x),T^{i+1}(x))\leq \varphi(T(x)) - $(something non-negative) which is of course less or equal than $\varphi(T(x))$.

The fact that $\varphi$ is bounded follows from the fact that $T$ is a contraction map.