" " and " " refer to exactly the same sequence, just renumbered. For example, , gives the sequence 1, 1/2, 1/3, 1/4, ... The sequence gives 1/2, 1/3, 1/4, ..., exactly the same except for the missing "1". And that doesn't change the limit, 0.
So you have . Since square root is continuous, taking the limit on both sides, . If we call the limit A, A must satisfy . That's the equation you have.
Now, that is assuming the sequence has limit. You say you know that the sequence is increasing. If you can show it has an upper bound, then you know it converges.