Schwartz space is dense in $L^{p}$

Question

I am trying to prove that Schwartz space $\mathscr{S}(\mathbb{R}^{n})$ of rapid decreasing functions is dense in $L^{p}(\mathbb{R}^{n})$ using the hints given in this previous MSE post.

Given $n \in \mathbb{N}$, let $R_{n}$ be an open rectangle $R_{n} = (-n,n)\times \cdots \times (-n,n) \subset \mathbb{R}^{n}$, where $(-n,n)$ is the open interval $-n < x < n$. Let $\chi_{R_{n}}$ be the indicator function $\chi_{R_{n}}(x) = 1$ if $x \in R_{n}$ and zero otherwise. If $f \in L^{p}(\mathbb{R}^{n})$, then $f_{n} := \chi_{R_{n}}f$ converges pointwise to $f$ and by an application of the Dominated Convergence Theorem, $\|f-f_{n}\|_{L_{p}} \to 0$ as $n \to \infty$.

Now, let $t > 0$ and set $K_{t}(x) = (it)^{\frac{-n}{2}}e^{-\frac{\|x\|^{2}}{t}} \in \mathscr{S}(\mathbb{R}^{n})$. In addition, given $\epsilon > 0$, take $n \in \mathbb{N}$ such that $\|f - f_{n}\|_{L_{p}} < \epsilon$. Define $g_{t} := f_{n}*K_{t}$. This function belongs to $L^{p}(\mathbb{R}^{n})$ because of inequality $\|f*g\|_{L_{p}}\le \|f\|_{L_{p}}\|g\|_{L_{p}}$. It is also smooth because: $$(f_{n}*K_{t})(x) = \int_{\mathbb{R}^{n}} f(y)K_{t}(x+y)dy$$ and by exchanging the derivative with the integral, $f_{n}*K_{t}$ is smooth because $K_{t}$ is smooth. Actually, $f_{n}*K_{t} \in \mathscr{S}(\mathbb{R}^{n})$.

I need to prove that $\|g_{t}-f\|_{L_{p}} \to 0$ as $t \to 0$, which is the point I am stuck right now. By Minkowski integral inequality, one has: $$\|g_{t}-f\|_{L_{p}} \le \int_{\mathbb{R}^{n}}\bigg{[}\int_{\mathbb{R}^{n}}|f_{n}(x-y)-f(x)|^{p}dx\bigg{]}^{\frac{1}{p}}K_{t}(y)dy$$

and I don't know how to move forward. I Think I need to use the $\epsilon$ argument to estimate $|f_{n}(x-y)-f(x)|$, but these are not evaluated at the same point, so I am not sure how to do it. In addition, what is the argument behind the $t\to 0$ implying the remaining integral going to zero? Can someone please help me finish the proof? Thanks in advance!