$Z_i$. Assume that conditions of Definition \[def:limit\] are satisfied.

Let $X$ be an at most $n$-dimensional metric space. Let $f \colon X \to Z_1$. Let $A$ be a closed subset of $X$. Let $g_0 \colon A \to Z$ be a map such that $\pi_1 \circ g_0 = f_{|A}$ (a lift of $f$ on $A$).

Then there exists a map $g \colon X \to Z$ such that $g_{|A} = g_0$ ($g$ extends $g_0$).

Let $f_1 = f$. Applying Lemma \[lem:single lift\] for each $i>1$ we construct a map $f_i \colon X \to Z_i$ that satisfies the following conditions.

1.  $p_{i-1} \circ f_{i}$ is ${{\mathcal{F}_{i-1}}}$-close to $f_{i-1}$.

2.  $\pi_i \circ g_0 = f_i\mid_A$.

For each $k$ and each $m > k$ we let $$a_m^k=\pi^m_k\circ f_m: X \to Z_k.$$ By (i), we have ${{d_{\sup\nolimits}}}(f_m, p_m \circ f_{m+1}) \leq \operatorname{mesh}{{\mathcal{F}_{m}}}$. By (B) the short projection $\pi^m_k$ is $1$-Lipschitz, hence ${{d_{\sup\nolimits}}}(a^k_m, a^k_{m+1})
  = {{d_{\sup\nolimits}}}(\pi^m_k \circ f_m, \pi^m_k \circ p_m \circ f_{m+1}) \leq \operatorname{mesh}{{\mathcal{F}_{m}}}$. Therefore for $l > m$ we have $${{d_{\sup\nolimits}}}(a^k_l, a^k_m) \leq {{d_{\sup\nolimits}}}(a^k_l, a^k_{l-1}) + {{d_{\sup\nolimits}}}(a^k_{l-1}, a^k_{l-2})
    + \cdots + {{d_{\sup\nolimits}}}(a^k_{m+1}, a^k_m) \leq$$ $$\leq \operatorname{mesh}{{\mathcal{F}_{l-1}}} + \operatorname{mesh}{{\mathcal{F}_{l-2}}} + \cdots + \operatorname{mesh}{{\mathcal{F}_{m+1}}} + \operatorname{mesh}{{\mathcal{F}_{m}}}.$$ By (E), the sequence $a_m^k$ is uniformly convergent. Let $$a^k = 