The fact that on (0,1) means, basically, that if we find , our answer will be less than on that same interval. Thinking about the integral as the area under the curve, if , then the area under sec(x) on (0,1) must be greater than or equal to the area under .
So to find the lower bound, we really need to evaluate:
, which gives
evaluated from 0 to 1, which gives your solution of .
Hope this helps!