Proof of Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous

Each result cited is universally quantified over the data in its own statement.

If $Z=\emptyset$, every map from $Z$ to $\mathbb{R}$ is continuous on $Z$ vacuously by Continuous Map Between Metric Spaces, and there is nothing to prove. Assume therefore that $Z\ne\emptyset$ and fix $x_{0}\in Z$. Then Continuity of Sums and Products of Real-Valued Functions on a Metric Space applies with its metric space $(Z,d)$, with $A=Z$ and with $x=x_{0}$, and claim Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set of it says: if $f,g:Z\to\mathbb{R}$ are continuous on $Z$ and $c\in\mathbb{R}$, then the maps $z\mapsto f(z)+g(z)$ and $z\mapsto c\,f(z)$ are continuous on $Z$.

Enumerating $T$. Since $T$ is finite and nonempty, by Finite Set there is a natural number $m$ such that $T$ has $m$ elements, that is, there is a bijection $\theta:[m]\to T$. For $z\in Z$ let $a^{z}:[m]\to\mathbb{R}$ be the map with values $a^{z}_{k}=b_{\theta(k)}\,f_{\theta(k)}(z)$. By Sum over a Finite Index Set, applied to the map $T\to\mathbb{R}$, $t\mapsto b_{t}\,f_{t}(z)$, and to the bijection $\theta$,

$$\sum_{t\in T}b_{t}\,f_{t}(z)=\sum_{k=1}^{m}a^{z}_{k}\qquad\text{for every }z\in Z. \tag{1}$$

For $j\in[m]$ let $g_{j}:Z\to\mathbb{R}$ be the map $g_{j}(z)=\sum_{k=1}^{j}a^{z}_{k}$, the finite sum of $a^{z}_{1},\dots,a^{z}_{j}$.

Induction. Let $I$ be the set of natural numbers $n$ such that, if $n\le m$, then $g_{n}$ is continuous on $Z$. We show that $I$ satisfies the two hypotheses of Principle of Induction for the Natural Numbers.

$1\in I$: we have $1\in[m]$ by Properties of the Order on the Natural Numbers §least, and by claim 1 of Properties of Finite Sums, applied to $a^{z}$, we have $g_{1}(z)=a^{z}_{1}=b_{\theta(1)}\,f_{\theta(1)}(z)$ for every $z\in Z$. Since $f_{\theta(1)}$ is continuous on $Z$, $g_{1}$ is continuous on $Z$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, with $f=g=f_{\theta(1)}$ and $c=b_{\theta(1)}$.

$S(n)\in I$ for every $n\in I$: let $n\in I$ and suppose $S(n)\le m$. By Properties of the Order on the Natural Numbers §successor, $n<S(n)$, so $n\le S(n)$ and then $n\le m$ by Properties of the Order on the Natural Numbers §basic; since $n\in I$, $g_{n}$ is continuous on $Z$. As $S(n)\in[m]$, claim 1 of Properties of Finite Sums, applied to $a^{z}$, gives

$$g_{S(n)}(z)=g_{n}(z)+a^{z}_{S(n)}=g_{n}(z)+b_{\theta(S(n))}\,f_{\theta(S(n))}(z)\qquad\text{for every }z\in Z.$$

The map $z\mapsto b_{\theta(S(n))}\,f_{\theta(S(n))}(z)$ is continuous on $Z$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, because $f_{\theta(S(n))}$ is; hence $g_{S(n)}$, the sum of two maps continuous on $Z$, is continuous on $Z$ by the same claim. Thus $S(n)\in I$.

By Principle of Induction for the Natural Numbers, $I=\mathbb{N}$. In particular $m\in I$, and $m\le m$ by Properties of the Order on the Natural Numbers §basic, so $g_{m}$ is continuous on $Z$. By (1), $g_{m}$ is the map $z\mapsto\sum_{t\in T}b_{t}\,f_{t}(z)$, which proves Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous.