To prove the inequality 1 < 1/5 + 1/6 + 1/7 + ..... + 1/17 < 2, we can simplify the expression and compare it to the bounds.

First, let's simplify the expression:

1/5 + 1/6 + 1/7 + ..... + 1/17 = 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + 1/17

Since the denominators are increasing, we can see that each term is less than 1/5. Therefore, the sum of these terms is less than 13*(1/5) = 13/5 = 2.6.

So, 1 < 1/5 + 1/6 + 1/7 + ..... + 1/17 < 2.6

Therefore, the inequality 1 < 1/5 + 1/6 + 1/7 + ..... + 1/17 < 2 is proven.