Use the definition of limit

Use the definition of limit to show that lim(n→∞) (n+1)/(n+2) = 1

i'm not sure what is definition of limit,

but this is my way of doing it,

(n+1) / (n+2)

= (n+2-1) / (n+2)

= 1 - 1/(n+2)

= 1 - 0

= 1

can anyone tell me if i did something wrong, thanks

Given ε > 0, we need to find a positive integer N such that
|(n+1)/(n+2) - 1| < ε for all n > N.

To show this, note that
|(n+1)/(n+2) - 1|
= |(n+1)/(n+2) - (n+2)/(n+2)|
= 1/(n+2)
< 1/n.

So, given ε > 0, choose N so that 1/N < ε <==> N > 1/ε.
Then for all n > N, we have |(n+1)/(n+2) - 1| < 1/n < 1/N < ε, as required.

I hope this helps!