First of all, you know that a fraction is smaller than 1 if the top number is smaller than the bottom. It works both ways, meaning that if the top number is larger than the bottom number, the fraction is greater than 1.
So, for example, a fraction less than 1 can be expressed as
(x-1)/x
because the top number is smaller than the bottom number.
The reciprocal can be found by flipping the top and the bottom, so the reciprocal is
x/(x-1)
x is always greater than x-1, so in the reciprocal, the top number will be larger than the bottom number. That means it will always be greater than 1.
Hope this helps!
(Sorry if this is confusing)