I know that the sequence
$\displaystyle (1+kx)^\frac{1}{k},$
where the sequence $\{k_i\}$ converges to zero, converges to $e^x$.
I also know the sequence is increasing. How does one show this is increasing? I am interested in neat ways of establishing the inequality,
$(1+ax)^\frac{1}{a} \ge (1+bx)^\frac{1}{b}$ if $b \ge a$
rather than the sequence itself.