I've been reviewing mathematical induction and I don't really get how to prove inequalities by induction. I've been working on this problem for a while..

so far I've proved n=1 for the base, then I assume n=k.

then, I use n=(k+1)..

this is where i get confused, I did this..

I multiplied by 2 because you can rewrite

as

but i'm not sure if multiplying by two on the other side is right since it's an inequality and not an equation like what i've been working with up until now. I would really appreciate help with trying to solve this