one often omits a universal quantifier. e.g. you did not object to "for all x" being omitted in statement (1) of the lemma above.
so since you point it out, let me balance both statements. i.e. they might reasonably read:
either:
lemma: the following are equivalent: (the universe of all variables is the real numbers, except for f which denotes a function);
1) for every e>0 there is a d>0 such that for all x, |x-x0| < d implies |f(x)-L| < e.
2) for every sequence {xn} converging to x0, the sequence {f(xn)} converges to L.
or: sloppier:
lemma: the following are equivalent: (omitting both universal quantifiers on the x's)
1) for every e>0 there is a d>0 such that |x-x0| < d implies |f(x)-L| < e.