$g_{n}=h_{n-1}^{-1}\circ h_{n}\in G_n$. Then $h=h_{m}={\operatornamewithlimits{\overrightarrow{\textstyle\prod}}}_{n\le m}g_n$. It remains to prove that each homeomorphism $g_n$ belongs to the neighborhood $U_n$. This will follow as soon as we check that $|g_n(x)-x|<{\varepsilon}_n$ for any $x\ne g_n(x)$.

Since $h_{n-1}|{\mathbb I}_{n-2}=h_{n}|{\mathbb I}_{n-2}=h|{\mathbb I}_{n-2}$, we conclude that $x\in {\mathbb I}_{n}\setminus{\mathbb I}_{n-2}$. It follows from $h_{n}\in G_n\cap B(1/2)$ that the point $y=h_{n}(x)$ belongs to the set ${\mathbb I}_{n}\setminus{\mathbb I}_{n-3}$. Since $h_{n-1}\in G_{n-1}\cap B(1/2)$, the point $z=h_{n-1}^{-1}(y)$ belongs to ${\mathbb I}_{n}\setminus{\mathbb I}_{n-4}$.

We claim that $$\label{H1}
|h_{n-1}(z)-z|<{\varepsilon}_{n}/2.$$ If $z\in {\mathbb I}_{n-2}\setminus{\mathbb I}_{n-4}$, then $|h_{n-1}(z)-z|=|h(z)-z|<{\varepsilon}(z)\le {\varepsilon}_{n}/2$ by the condition (\[E1\]) from the definition of the function ${\varepsilon}$. If $z\in {\mathbb I}_{n-1}\setminus{\mathbb I}_{n-2}$, then the linearity of $h_{n-1}$ on the two intervals composing the set ${\mathbb I}_{n-1}\setminus{\mathbb I}_{n-2}$ implies that $$|h_{n-1}(z)-z|\le\max_{t\in\partial{\mathbb I}_{n-2}}|h(t)-t|<\max_{t\in\partial{\mathbb I}_{n-2}}{\varepsilon}(t)\le
{\varepsilon}_{n}/2$$by the definition of the function ${\varepsilon}$. Here $\partial{\mathbb I}_{k}=\{k,-k\}$ stands for the boundary of the interval ${\mathbb I}_{k}=[-k,k]$ in ${\mathbb R}$.

By a similar argument we can prove the inequality $$\label{H2}
|h_{n}(x)-x|<{\varepsilon}_{n}/2.$$

Unifying (\[H1\]) and (\[H2\]) we obtain the desired inequality: $$\begin{aligned}
|g_{n}(x)-x|&=|h_{n-1}^{-1}\circ h_{n}(x)-x|\le |h_{n-1}^{-1}\circ h_{n}(x)-h_{n}(x)|+|h_{n}(x)-x|=\\
&=|z-h_{n-1}(z)|+|h_{n}(x)-x|<\frac12{\varepsilon}_{n}+\frac12{\varepsilon}_n={\varepsilon}_n.
\end{aligned}$$

Next, we present an example of a tower $(G_n)_{n\in{\omega}}$ that satisfies ${\mathrm{PTA}}$ but is not (bi-) balanced.

Let $(e_n)_{n\in{\omega}}$ be an orthonormal basis of the separable Hilbert space 