Using the Squeezing Theorem of Limits in Mathematica

I wanted to compute the following limit

Limit[ρ, {α -> 0}, Direction -> "FromAbove", Assumptions -> 1 <= ρ <= 1 + α]

clearly this has the answer $\alpha$ by using the squeezing theorem. However, I don't know how to get the right answer from Mathematica.

• I think this topic is too hard for current CASes. For example, Limit[[Rho][[Alpha]], [Alpha] -> 0, Direction -> "FromAbove", Assumptions -> ForAll[[Alpha], [Alpha] >= 0, 1 <= [Rho][[Alpha]] && [Rho][[Alpha]] <= 1 + [Alpha]]] fails. – user64494 Jun 2 '18 at 17:23
• @user64494 But Reduce[ForAll[\[Alpha], \[Alpha] >= 0, 1 <= \[Rho] && \[Rho] <= 1 + \[Alpha]]] works just fine... – Henrik Schumacher Jun 2 '18 at 17:55
• @Henrik Schumakher: So what? The question is unclearly formulated. The command Limit[[Rho], [Alpha] -> 0, Direction -> "FromAbove", Assumptions -> ForAll[[Alpha], [Alpha] >= 0, 1 <= [Rho] && [Rho] <= 1 + [Alpha]]] produces [Rho] instead of 1. – user64494 Jun 2 '18 at 18:07