Question regarding this answer to a limit problem

Question

This is regarding this answer which sought to address this problem:

Prove that $\lim_{n\to\infty} a_n=0$ given that $|x| \lt 1$.

I perfectly understand the answer except the last few lines which solves the problem using Bernoulli's inequality. I think it should be something like this:
$(1+t)^n \gt 1+nt \ge (1+2mt)$
But how is $(1+t)^n \gt (1+mt)^2$ ? $a \gt b $ and $c \gt b$ do not imply $a \gt c$!

Answer 1

If $$(1+t)^m \ge 1+mt$$

Note that we have $\frac{n}{2} \ge m$

Hence we have $(1+t)^{\frac{n}{2}} \ge (1+t)^m \ge (1+mt)$

Now we square everything and we conclude that

$$(1+t)^n \ge (1+mt)^2$$