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In http://www.uni-hamburg.de/fachbereiche-einrichtungen/fb03/iwwt/makro/slides3.pdf, you can see New Keynesian IS equation being solved forward to yield a formula for output gap (page 40 of the pdf slides). I do get that it is based on output gap becoming zero at infinity. But exactly what condition is this justification based on? Is it $\lim_{T \to \infty}E_t[\tilde{y_T}] = 0$? Or $E_t[\lim_{T \to \infty}\tilde{y_T}] = 0$ Or other conditions?

Also when getting the solution for new keyneisan models as done in later pages of the pdf (especially Blanchard-Kahn in page 44), if it is true that the above output gap formula depends on $\lim_{T \to \infty}E_t[\tilde{y_T}] = 0$ or $E_t[\lim_{T \to \infty}\tilde{y_T}] = 0$, does the solution mechanism also depend on these?

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