Is it possible for a sequence in $L^p$ to converge to an element of $L^q$ ($q>p$) w.r.t. the $q$-norm?

Question

Suppose that $1 \leq p < q \leq \infty$ and that $f_n$ is a sequence in $L^p(\mathbb{R})$.

If $f_n$ is a Cauchy sequence with respect to the metric induced by the $L^p$ norm, then (I believe that) it must converge to a function in $L^p$, because $L^p$ is a Banach space and therefore a complete metric space.

But is it possible for the following to all hold simultaneously?

1. $f_n$ is not a Cauchy sequence with respect to the $L^p$ norm.
2. $f_n$ is a Cauchy sequence with respect to the $L^q$ norm.
3. $f_n$ converges to an element of $L^q(\mathbb{R}) \setminus L^p(\mathbb{R})$ with respect to the $L^q$ topology.

For example, is it possible for a sequence of functions in $L^2(\mathbb{R})$ to diverge w.r.t the $L^2$ metric but converge to a function in $L^3(\mathbb{R})$ w.r.t. the $L^3$ metric?

I'm imagining a situation where (loosely speaking) if $M$ and $N$ are any sufficiently large integers then $f_M$ and $f_N$ have tails that are different enough that $$\int_{x_0}^\infty |f_M(x) - f_N(x)|^p\ dx = \infty,$$ but similar enough that $$\int_{x_0}^\infty |f_M(x) - f_N(x)|^q\ dx < \epsilon$$ for any $\epsilon > 0$.

The reason I ask is that I believe that

1. $L^p$ spaces are Banach spaces for all $p \in (0,\infty]$,
2. Schwarz space $S(\mathbb{R})$ is a dense subspace of $L^p(\mathbb{R})$ for all $p \in [1,\infty)$.
3. $L^p(\mathbb{R}) \neq L^q(\mathbb{R})$ if $p \neq q$.

From these three facts, I gather that the limit points of Schwartz space $S(\mathbb{R})$ must depend on which $L^p$ topology is being inherited; a sequence in $S(\mathbb{R})$ that converges with respect to the $L^p$ metric must converge to a function in $L^p(\mathbb{R})$. Is this correct?