It says: Show that the sum from k=1 to 1000 of 1/(1+k^2) is smaller or equal to 1000.
Now I see that they want you to use the 5th property of summation in the page you showed! Since you didn't mention an inequality, I didn't consider that as being relevant.
Now let's think about why you left out
the most important part of the problem.
I imagine that you assumed that if they ask you to
show that the sum is less than a certain number, they expect you to
find the sum in order to do that. In most school math, you are only given problems that can be solved that directly. But if you are in calculus and studying series, you are old enough to know the truth:
Most equations can't be solved, and most problems can't be answered exactly! You first learn only the problems that can be solved easily, and only later learn about real life, if ever.
Such naivete is the main reason we ask everyone to show the
entire problem they are working on, not just the piece they
think is important: You don't know yet what matters, or how wrong your initial thinking may be! That's true on any topic that is new to you.