Mathematical Proof as to why does the total revenue increase when the prices are increased for an inelastic product?
(I do not agree that this is a duplicate of the answer proposed by other members here).  

Total Revenue is

$$TR = Q(P)\cdot P$$

The derivative with respect to price is 

$$\frac{\partial TR}{\partial P} = Q'(P)\cdot P + Q(P)$$

For this to be positive (and so total revenue to increase with price), we require 

$$\frac{\partial TR}{\partial P} &gt; 0 \implies  Q'(P)\cdot P + Q(P) &gt;0 $$

$$\implies \frac{Q'(P)\cdot P}{Q(P)} &gt; -1$$

The left-hand-side is the price elasticity of demand. Now, greater than minus unity means smaller than unity in absolute terms. And it is in that sense that demand at the region is "price inelastic" (the minus sign indicating direction of movement).