Let $\alpha, \beta, \delta$ be ordinals. If $|\alpha|\leq\aleph_\beta$, $$\aleph_{\delta+\alpha}^{\aleph_\beta}=\aleph_{\delta+\alpha}^{|\alpha|}\aleph_{\delta}^{\aleph_\beta}.$$ (You can assume Hausdorff´s formula).
I´ve been trying to prove this problem for several weeks and still got no solution. According to the hint given, it must be proved by the Second Version of the Transfinite Induction Principle (Hrbacek & Jech, 1999) on $\alpha$, so I started and got stuck on the $\alpha$ limit ordinal case.
Any help you can give me will be aprreciated.