I am proving some inequality and one of the steps i took is that i wrote $n+1 leq n+1$ is it correct mathematically to write such thing ?

Note :

i was try to prove that $2^{n-1} leq n! leq n^{n} space space forall n in mathbb{N^*}…P_n $

so i did it by induction and i got to $2^{n-1} leq n! leq (n+1)^n$

then i wrote that $2 leq n+1 leq n+1$ and i multiplied the inequalities side by side to get to $P_{n+1}$