sum (n+1)/(n+2) as n goes from 1 to infinity

my solution:

let {Xn}= (n+1) /(n+2)

for large n , the series is approx. to 1

where sum 1 is divergent.

so i am going to show that {Xn} >= C .1 where C is a constant

in fact, n+1>n and n+2=< 3n

so Xn=(n+1)/(n=2)> n/3n=(1/3) . 1

since sum 1 diverges, so does sum (1/3).1

so by comparison test, sum Xn is divergent.

is my proof correct? anybody help me? many thanks~!