\(\eqalign{
& {{{n^2}} \over {{n^2} + 1}} - {1 \over 2} = {{2n - ({n^2} + 1)} \over {2({n^2} + 1)}} = {{ - {{(n - 1)}^2}} \over {2({n^2} + 1)}} \le 0;\,\,\forall n \in {N^*} \cr
& \Rightarrow {n \over {{n^2} + 1}} < {1 \over 2};\,\,\forall n \in {N^*} \cr
& {{{n^2} + 1} \over {2n}} - 1 = {{{n^2} + 1 - 2n} \over {2n}} = {{{{(n - 1)}^2}} \over {2n}} \ge 0;\,\,\forall n \in N* \cr
& \Rightarrow {{{n^2} + 1} \over {2n}} \ge 1;\,\,\forall n \in {N^*} \cr} \)
