Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $mathbb{R}[X]_{leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $mathbb{R}[X]$ of variety $X$.

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Definition: A polynomial $f$ is $d$-SOS if there exist $g_{1}, dots, g_{k} in mathbb{R}[X]_{d}$ such that $$f = g^{2}_{1} + cdots + g^{2}_{k}$$

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Question: Suppose that you have a finite set $Xsubset mathbb{R}^{n}$ and a sequence ${f_{i}}_{iin mathbb{N}}$ of $d$-SOS polynomials in $mathbb{R}[X]_{leq 2d}$, then is $$f := lim_{i in mathbb{N}} f_{i}$$ also $d$-SOS?

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For me, it’s clear that any non-negative polynomial in $X$, as it is finite, is sum of squares (SOS), but for me it is not clear why the limit has degree at most $2d$. Is it true? Why?