Prove that the sequence converges

[demonelite123]

x_n = (n^2 / (n^2+1) , 1/sqrt(n)). prove that this sequence converges and find the limit.

so as n approaches infinite it is clear that x_n approaches (1,0). so using the definition of a convergent sequence, i pick an epsilon > 0 and i have to find some N such that when n>N, sqrt( 1/(n^2 + 1)^2 + 1/n ) < epsilon.

i tried fiddling with the inequality for a while but i can't seem to get it into a form where i can flip the inequality around and say n > (some function of epsilon). i tried combining the fraction into [(n^2+1)^2 + n] / [n(n^2+1)^2] and expanding it but i seem to be getting nowhere. can someone maybe give me a few hints or pushes in the right direction? thanks.

these type of proofs seem to be mostly exercises in manipulating inequalities which i am still inexperienced at.

 

[LCKurtz]

If you are trying to make something like

$$\sqrt{a^2 + b^2}$$

small by making positive numbers a and b small, think about getting each of a and b less than 1. Then you will know a² < a. Then if you let m = max{a,b}, how big can the square root expression be, and can you make it small by taking a and b additionally smaller than one?