Let

#sqrt(n+1)+sqrt(n-1)" is rational and can be expressed by " p/q #

where p and q prime to each other and#" "q!=0#

So

#sqrt(n+1)+sqrt(n-1)=p/q.........(1)#

Inverting (1) we get

#1/(sqrt(n+1)+sqrt(n-1))=q/p#

#=>(sqrt(n+1)-sqrt(n-1))/((sqrt(n+1)+sqrt(n-1))(sqrt(n+1)-sqrt(n-1)))=q/p#

#=>(sqrt(n+1)-sqrt(n-1))/2=q/p#

#=>(sqrt(n+1)-sqrt(n-1))=(2q)/p.....(2)#

Adding (1) and (2) we get

#2sqrt(n+1)=p/q+(2q)/p#

#=>sqrt(n+1)=(p^2+2q^2)/(2pq).....(3)#

Similarly subtracting (2) from (1) we get

#=>sqrt(n-1)=(p^2-2q^2)/(2pq).....(4)#

Since p and q are integers then eqution (3) and equation(4) reveal

that both#" "sqrt(n+1) and sqrt(n-1)#

#color (blue)(" are rational as their RHS rational")#

So both #(n+1) and (n-1) " will be perfect square"#

Their difference becomes #(n+1)-(n-1)=2#

But we know any two perfect square differ by at least by 3

Hence it can be inferred that there is no positive integer for which

#sqrt(n+1)+sqrt(n-1)" is rational"#