difficult question on monotonic function

你是不是想得太複雜了?我會這樣想
prove a STRICTLY monotonic increasing function is injective.

if the statement is not true, then there exist x1 and x2 (x1 <> x2) such that f( x1)=f( x2)
However this is impossible since according to the definition of STRICTLY monotonic increasing function, if x1 <> x2 then either f( x1)>f( x2) or f( x1)<f( x2). So there is a contradiction and we prove that a STRICTLY monotonic increasing function is injective