I'm reading through Terence Tao's Real Analysis II, and he made a seemingly off-hand comment that made me pause and think.
"If $(x^{(n)})_{n=m}^\infty$ converges to $x$, then $(x^{(n)})_{n=m'}^\infty$ also converges to $x$ for any $m'\geq m$." In his notation, $(x^{(n)})_{n=m}^\infty$ is a sequence that starts at the $m$th index and is indexed by $n$.
Clearly if this statement also held if $m'<m$, then he would've mentioned it, and yet I can't think of a solid reason that it couldn't hold. Does it have something to do with Riemann's rearrangement theorem?
Simply, adding or removing finite number of elements to/from a sequence doesn't affect its convergence.
So, it does hold with $m'<m$ as well (provided those elements are defined).