If $f_n$ is a sequence of functions over $[0, \pi/2]$ given by

$$f_n(x) = \frac {nx} {1+n\sin(x)},$$

then how would I go about proving that $f_n$ does not converge uniformly to a function $f$ on $[0, \pi/2]$?


Question: Water Supply pressure
Answer: Given that 20 meters of head only equates to 1.96 Bar (1m elevation = 0.0980413943 Bar), no pipe alone will do this - you'll need a pump, or you'll need to accept less than 2 Bar as all the pressure you have available.

Then you need to determine the flow rate you wish to operate at for sizing the pipe - you'll get 1.96 bar with any size pipe and no flow (static head) but as you actually use water the pressure will drop (dynamic head) in relation to to pipe size and rate of flow. That is essentially friction in the pipe resisting flow and reducing the effective head. Calculators are available and the formulas for that are adequately complex that using one is suggested. Aside from size of pipe and flow rate, pipe material enters into it as some types are smoother inside and thus have less friction for a given size of pipe than other types.

For example, if you were sizing for 12 l/min, 25mm plastic pipe would have about 2.7 m dynamic head, leaving you with 17.3m head pressure (delivery pressure = 1.7 Bar), and 50 mm pipe would only lose 0.0925m at that flow rate (delivery pressure = 1.9 Bar). 

Bump the flow up to 120 l/min and the 50 mm pipe costs 6.55m head (delivery pressure = 1.3 Bar) - but the 25mm pipe cannot provide that flow rate at all with 20m head - it tops out at about 35 l/min with no pressure at the house end (20 m static head - 20m dynamic head; delivery pressure = 0 Bar)