Convergence in norm equivalent formulations

Question

I have a short question:
For $g_n, g \in \mathcal L^p(\mathbb R^n, \mathcal B(\mathbb R^n), \lambda _n)$ is $$\lim_{n\to \infty} \lVert g_n -g \rVert_p = 0 \Longleftrightarrow \lim_{n\to \infty} \lVert g_n\rVert _p = \lVert g \rVert_p?$$ In a proof of a problem I want to show $g_n \to g$ w.r.t. $\lVert \cdot \rVert _p$ using dominated convergence but showing the right hand side above seems to be easier as you automatically have a integrable bound.

Answer 1

No its not. Let $g_n=1_{B_r(0)}$ and $g=-1_{B_r(0)}$, then $\|g_n\|_p=\|g\|_p$ but $g_n$ does not converge to $g$ in $L^p$.