What facts are you allowed to use? Certainly, defining has the properties in (1) doesn't it? And, if so, then you have shown that that function, at least, has those properties.

For 2, assume y> 0 and look at exp(x+ y)= exp(x)*exp(y)> exp(x)(1+y). Do you see how that shows that exp(x+y)> exp(x)? Do you see why that says that exp(x) is an increasing function?