PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$
Use mathematical induction to prove this. Here is my answer but I stuck at certain point.
Base Case: n=1 $$(1+x)^1 \ge 1+x $$ True ,
Induction Case: n=k assume $$(1+x)^k \ge 1+kx $$ n=k+1 $$ (1+x)^k+1 \ge 1+(k+1)x $$ $$(1+x)^k *(1+x) \ge 1+ kx+ x $$
Stuck!!!
Bernoulli's Inequality is proven for integer exponents in this answer by induction and then expanded to rational exponents in this answer.
This is the Bernoulli Inequality, whose proof (by induction) can be found on Wikipedia.