Data Sufficiency

I was just working my way through the DS questions in the 2nd Edition Quant Review but I can´t wrap my head around the answer explanation of DS Question Number 107.
It would take too long to write down the whole answer explanation but maybe someone of you has the book and could explain the solution to me in other words?
Thanks a lot in advance!

The sequence s1, s2, s3,.....sn,...is such that Sn= (1/n) - (1/(n+1)) for all integers n>=1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?

1) k > 10
2) k < 19

WRITE IT OUT UNTIL YOU SEE A PATTERN.

S(1) = 1 - 1/2
S(2) = 1/2 - 1/3
S(3) = 1/3 - 1/4
etc.

Sum of the first 3 terms:
S(1) + S(2) + S(3) = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) = 1 - 1/4 = 3/4.

When the terms are added, every value except the first and the last CANCELS out.
The result is that, when n=3, the sum = n/(n+1) = 3/4.
Thus, when n=k, we can DEDUCE that the sum = k/(k+1).

Question rephrased: Is k/(k+1) > 9/10?

Statement 1: k>10.
If k=11, then the sum = k/(k+1) = 11/12, which is GREATER than 9/10.
If k=12, then the sum = k/(k+1) = 12/13, which is GREATER than both 9/10 and the preceding sum, 11/12.

As k increases, so does the sum.
Since the LEAST possible sum here = 11/12, the sum will always be GREATER than 9/10.
SUFFICIENT.

Statement 2: k<19.
As shown in statement 1, if k=11, then the sum is GREATER than 9/10.
If k=1, then the sum = k/(k+1) = 1/2, which is LESS than 9/10.

Since the sum in the first case is GREATER than 9/10 and the sum in the second case is LESS than 9/10, INSUFFICIENT.

The correct answer is A.