Why do you write things like {{f}_{n}} in TeX instead of just f_n?  Or \underset{n\to\infty}{\mathop{\lim}} instead of \lim_{n\to\infty} ?  \lim is already a standard operator name, and when it appears in a "displayed" as opposed to "inline" setting, its subscripts appear directly below it, without the need to use "underset".  (Similarly with \sup.)