Consider a function $ \gamma(t) = (f\_1(t), f\_2(t), ... , f\_k(t))$ for $t \in [a,b]$. So, $f\_i$ are real-valued functions on $[a,b]$. Then $\gamma$ is continuous, i.e. a path, if and only if each $f\_i$ is continuous.

Show that if the functions $f\_i$ are uniformly continuous, then so is $\gamma$.

Can anyone give me a tip on where to start for this problem?