Let {$\displaystyle s_n$(x)} be a sequences of functions which converges uniformly to f(x) on the intervals [a,b] and [b,c]. Prove that the sequence {$\displaystyle s_n$(x)} converges uniformly on [a,c] to f(x).
Imagine that you supervise two teams who install pre-assembled bridge sections. Each team moves their section closer and closer to where it should be. You can tell each team a distance (say, 1 meter), and the team will notify you when their section is within this distance from the target.
Now suppose that your supervisor requires the same from you: he/she tells you a distance and you have to notify him/her when both sections of the bridge are within this distance from the target. Upon receiving a distance, what is your strategy?