0 < ( n + 1 ) − n < ( n + 1 ) {\displaystyle 0<{\sqrt {(n+1)}}-{\sqrt {n}}<{\sqrt {(n+1)}}}
0 < ( ( n + 1 ) − n ) ( n + 1 + n ) < n + 1 ( n + 1 + n ) {\displaystyle 0<({\sqrt {(n+1)}}-{\sqrt {n}})({\sqrt {n+1}}+{\sqrt {n}})<{\sqrt {n+1}}({\sqrt {n+1}}+{\sqrt {n}})}
0 < 1 < n + 1 + ( n + 1 ) 1 / 2 {\displaystyle 0<1<n+1+(n+1)^{1/2}}
0 < 1 / ( n 1 / 2 ) < n + 1 + ( n ( n + 1 ) ) 1 / 2 ( n 1 / 2 ) < n + 1 + ( n ( n + 1 ) ) 1 / 2 {\displaystyle 0<1/(n^{1/2})<{\frac {n+1+(n(n+1))^{1/2}}{(n^{1/2})}}<n+1+(n(n+1))^{1/2}}
n 1 / 2 − ( n + 1 ) 1 / 2 < n < n + 1 < n + 1 + ( ( n + 1 ) n ) 1 / 2 {\displaystyle n^{1/2}-(n+1)^{1/2}<n<n+1<n+1+((n+1)n)^{1/2}}
1 / ( n ) 1 / 2 < ( n + 1 + ( ( n + 1 ) n ) 1 / 2 ) / ( n 1 / 2 ) {\displaystyle 1/(n)^{1/2}<(n+1+((n+1)n)^{1/2})/(n^{1/2})}
1 / ( n ) 1 / 2 ≤ ( n ) 1 / 2 − ( n + 1 ) 1 / 2 n 1 / 2 < ( n + 1 + ( ( n + 1 ) n ) 1 / 2 ) n 1 / 2 {\displaystyle 1/(n)^{1/2}\leq {\frac {(n)^{1/2}-(n+1)^{1/2}}{n^{1/2}}}<{\frac {(n+1+((n+1)n)^{1/2})}{n^{1/2}}}}
1 / ( n ) 1 / 2 < 2 ( ( n ) 1 / 2 − ( n + 1 ) 1 / 2 ) {\displaystyle 1/(n)^{1/2}<2((n)^{1/2}-(n+1)^{1/2})}