Determining convergence with taking the limit of a function greater than it Why does the Sandwhich Theorem have to be used here? Why couldn't we just test for the result of the right side of the inequality and say that $a_n$ converges, since the function greater than itself converges?


You may have confused with this correct theorem (called comparison test):
If $\sum b_n$ converges and $0\leq a_n\leq b_n$ then $\sum a_n$ also converges.

What you mentioned: " $a_n$ converges, if the function greater than itself converges" is not correct:
Example: $a_n=(-1)^n\leq 2$, and 2 being a constant sequence surely converges, but $a_n$ does not converge.
