That's the wrong start and I think this will become very technical.
We need to show that ##H_1\otimes H_2## is complete. This means we need to show that every converging sequence, or Cauchy sequence in it, converges to a point in ##H_1\otimes H_2##.
Now, ##v_1\otimes v_2## is not a typical element in ##H_1\otimes H_2## so we may not assume that elements look like that. A typical element is ##\sum_{k\in \mathbb{N}} c_k v_1^{(k)}\otimes v_2^{(k)}## with only finitely many scalar factors ##c_k \neq 0.## That's why I introduced them. We can write the sum without the ##c_k## but then it becomes more difficult to say that the sum is finite although we sum over potentially infinite bases.
A sequence, therefore, looks like
$$
\left(\sum_{k\in \mathbb{N}} c_{k,n} v_1^{(k,n)}\otimes v_2^{(k,n)}\right)_{n\in \mathbb{N}}
$$
If this is a Cauchy sequence, then the difference between two sequence members becomes as small as we like if we chose the indices high enough, i.e.
\begin{align*}
\left\|\sum_{k\in \mathbb{N}} c_{k,n} v_1^{(k,n)}\otimes v_2^{(k,n)}\, - \,\sum_{k\in \mathbb{N}} c_{k,m} v_1^{(k,m)}\otimes v_2^{(k,m)}\right\| <\varepsilon
\end{align*}
Since only finitely many ##c_{k,n}## and ##c_{k,m}## are different from zero, we can write this difference as
$$
\left\|\sum_{p\in \mathbb{N}} c'_{p} v_1^{(p)}\otimes v_2^{(p)}\right\|<\varepsilon
$$
with new coefficients and over all dyads that occur in either of the previous sums. Now to the crucial part: how is the norm defined? It is induced by the inner product, so the question is: what is the inner product in the tensor space? The answer is
\begin{align*}
\left\|\sum_{p\in \mathbb{N}} c'_{p} v_1^{(p)}\otimes v_2^{(p)}\right\|&=\ldots\\
&=\ldots \\
&=\ldots\\
&= \sqrt{\sum_{r\in \mathbb{N}} |c'_r|\cdot \langle v_1^{(r)}\,|\, v_1^{(r)}\rangle \cdot \langle v_2^{(r)}\,|\,v_2^{(r)} \rangle} < \varepsilon
\end{align*}
where the dots represent a lot of distributive multiplications, re-arrangement of the dyads, and re-indexing. But all factors in the sum under the root are positive, so they are all as small as we want. This means that the ##\sqrt{ \sum_p c'_p \langle v_1^{(p)} \,|\, v_1^{(p)} \rangle} ## and ##\sqrt{\sum_q c'_q\langle v_2^{(q)} \,|\,v_2^{(q)}\rangle }## are Cauchy sequences which converge in ##H_1##, resp. ##H_2,## say with limits ##L_1## and ##L_2.##
Finally, we have to go all the steps backward and show that
\begin{align*}
\left\|\sum_{p\in \mathbb{N}} c'_{p} v_1^{(p)}\otimes v_2^{(p)}\; - \; L_1\otimes L_2\right\| <\varepsilon
\end{align*}
That is the plan. However, I'd rather solve a 1,000-piece puzzle than fill all of the above with the correct epsilontic and all correct indices. Or call for a physicist to do some voodoo with all the indices.