Your base case is incorrectly worked out to start with.

The sum from i=1 to i=2 is 1/3+1/7=10/21

which is <1 so the base case is ok.

You have the P(k) proposition as

Then your P(k+1) proposition is

Then you try to prove that IF P(k) is true THEN P(k+1) will be true.

Do this by using the fact that if Pk) is true,

P(k+1) LHS will be less than or equal to

and hopefully this will be

Therefore we can ask if