In functional analysis, many properties of certain spaces are normally derived from taking a pointwise limit out of the norm, i.e.

$lvert lvert x rvert rvert=limlimits_{nto infty}lvert lvert x_{n} rvert rvert$ $(*)$.

The normal justification for this is that

$lvert lvert cdot rvert rvert: X to mathbb R$ is a continuous function wrt to the norm $lvert lvert cdot rvert rvert$ by the reverse triangle inequality.

Note that we merely used the definition of a norm to obtain continuity. In particular, $lvert lvert cdot rvert rvert_{L^{p}}$ is continuous.

However, when we arrive at $L^{p}$ spaces there are convergence theorems, like Dominated Convergence Theorem, Monotone Convergence Theorem and Fatou’s Lemma which of course imply that simply taking the limit out of the norm is not possible. Why is this not a contradiction to $(*)$?