We note that,

{y_n} is a postive sequence because,

n+1>n thus, sqrt(n+1)>sqrt(n) thus, sqrt(n+1)-sqrt(n)>0

Thus it has a lower bound.

The sequence is also strictly decreasing.

Because,

y_{n+1}<y_n

If and only if,

y_{n+1}-y_n<0

If and only if,

sqrt(n+2)-sqrt(n+1)-sqrt(n+1)+sqrt(n)<0

If and only if,

sqrt(n+2)-sqrt(n)<2sqrt(n+1)

Which is true (square both sides).

Thus, by Weierstrauss-Bolzano Theorem this sequence has a limit.