Suppose $y$ and {$y_n$}$_1^\infty$ $\in \ell^p$ s.t: $1 \leq p<\infty$, $y_n \rightarrow 0$ weakly.
Show:
$$\limsup||y_n+y||_p^p=||y||_p^p + \limsup||y_n||_p^p$$
The hint I'm given is to ''First assume $y$ has finite support''.
Attempting to follow the hint: suppose $y$ is supported on $\{1...N\}$.
Then we can write:
$$\limsup||y+y_n||_p^p=\limsup \sum_{i=1}^\infty |y_i+y_{n_i}|^p=\limsup( \sum_{i=1}^N|y_i+y_{n_i}|^p + \sum_{i=N+1}^\infty |y_{n_i}|^p)$$ = $$\limsup (\sum_1^N|y_{n_i}+y_i|^p - \sum_{i=1}^N |y_{n_i}|^p + ||y_{n_i}||_p^p)$$ Here I'm stuck. I cannot even imagine how I can use the weak convergence of $y_n$. It almost feels like I'm missing a basic property of summations. I can't seem to manipulate the first two terms at all.
EDIT: Using that $y_{n_i} \rightarrow 0$ I think I can say:
$$\limsup( \sum_{i=1}^N|y_i+y_{n_i}|^p + \sum_{i=N+1}^\infty |y_{n_i}|^p) \leq \limsup \sum_1^N |y_{n_i}+y_n|^p + \limsup||y_n||_p^p=\\ \limsup \sum_1^N |y_i|^p + \limsup||y_n||_p^p = ||y||_p^p + \limsup||y_n||_p^p$$
Extending this to a general $\ell^p$ vector is still very mysterious to me.