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

In the setting of The Real Numbers: Standing Notation and Background, let $(Z,d)$ be a metric space, let $\mathbb{R}$ carry the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line, let $T$ be a nonempty finite set, and for each $t\in T$ let $b_{t}\in\mathbb{R}$ and let $f_{t}:Z\to\mathbb{R}$ be continuous on $Z$ as a map from $(Z,d)$ to $(\mathbb{R},d_{\mathbb{R}})$. Sums over finite index sets are those of Sum over a Finite Index Set.

(Continuity)¶ The map $Z\to\mathbb{R}$, $z\mapsto\sum_{t\in T}b_{t}\,f_{t}(z)$, is continuous on $Z$.