On the limit of assessments that are not sequentially rational In game theory, in the attempt to define sequential equilibrium, for every tuple $b$ of behavioral strategies for each player, the corresponding belief system induced by $b$ is defined. A tuple $(b,μ)$ is called an assessment if $μ$ is the belief system induced by $b$.
Then, the criterion of an assessment being sequentially rational is defined.
Finally, sequential equilibrium is defined as an assessment $(b,μ)$ such that:


*

*$(b,μ)$ is sequentially rational.

*There exists a sequence of assessments $(b_i,μ_i)$ such that each $b_i$ assigns nonzero probabilities to all decisions and the sequence converges to $(b,μ)$.


My question is: can there be a sequence of assessments $(b_i,μ_i)$ satisfying condition number 2 and converging to a sequential equilibrium but with each element being not sequentially rational (but nevertheless converges to an assessment that is sequentially rational)?
EDIT:
Just in case the answer is YES, what if it is given that: each sequence of assessments satisfying condition number 2 -- and converging to an assessment $(b,μ)$ -- has $(b_i,μ_i)$ being not sequentially rational for all large enough $i$? Can then it be concluded that $(b,μ)$ is not sequentially rational?
I have a strong intuition that the answer must be YES, because if it isn't, then the definition of sequential equilibrium seems pointless.
 A: The answer is yes. Note that your condition 2 does not require that elements of the sequence be sequentially rational. 
Consider the following game: 

It is clear that the strategy profile $(b_1,b_2)=(A,L)$ and the belief system 
\begin{equation}
\mu_1(\varnothing)=1 
\qquad\qquad\qquad 
\mu_2(A)=1-\mu_2(B)=1
\end{equation}
form a sequential equilibrium. 
For any sequence of completely mixed strategies $(\sigma_1^n,\sigma_2^n)\to(b_1,b_2)$ with 
\begin{equation}
\sigma_1^n(A)=p_n
\quad\text{and}\quad 
\sigma_2^n(L)=q_n,\qquad p_n,q_n\in(0,1)\;\forall n,
\end{equation}
Player 2's consistent beliefs are $\mu_2^n(A)=p_n$. However, given $p_n\in(0,1)$, it is not sequentially rational for Player 2 to play $\sigma_2^n(L)=q_n\in(0,1)$; Player 2 should best respond with the pure strategy $L$, which is not feasible unless in the limit. 
Thus, the above example shows that, by forcing players to use proper mixed strategies, the sequence of assessments $(b_i,\mu_i)$ may prevent a player from playing the unique pure strategy best response to the other player's mixed strategy. As a result of such prevention, sequential rationality cannot be satisfied.
