The limit of I_n (nth iteration) is the same as I_{n+1} (n+1 th iteration).

Thus,

I_{n+1}=1+sqrt(I_n)

Thus,

If lim I_n= L then, lim I_{n+1}=L

Thus,

L=1+sqrt(L) (limit composition rule for sequnces)

Solve for L.

IMPORTANT. The fact that lim I_n exists was impervious to the proof. That I did not prove.

But the idea is to show that it is an increasing bounded sequence. (Use Wierestrass-Bolzano Theorom)