Mass Theory – Prove that $ mathcal B ( mathbb R) ^ infty subsetneq mathcal B ( mathbb R ^ infty) $.

I want to prove that $ mathcal B ( mathbb R) ^ infty subsetneq mathcal B ( mathbb R ^ infty) $, from where $ mathbb R ^ infty = prod_ {i = 1} ^ infty mathbb R $. $ mathcal B (A) $ is the Borel $ sigma – $Algebra of $ A $ and $ mathcal B ( mathbb R) ^ infty $ is the product $ sigma – $Algebra on $ mathbb R ^ infty $,

I do not really know how to go about it. I know that $ mathcal B ( mathbb R) ^ infty = sigma ( mathcal C) $ from where $ mathcal $ is the amount of cylinder, d. H. the amount of form $$ C_ {i_1, …, i_n} (B_1, …, B_n) = {x in mathbb R ^ infty mid x_ {i_1} in B_1, …, x_ {i_n } in B_n } $$ from where $ B_i $ Borel starts $ mathbb R $,

Now I know that $ mathcal B ( mathbb R ^ infty) $ is the $ sigma – $Algebra generated by open sets of $ mathbb R ^ infty $, A basis is given by $$ left { prod_ {i = 1} ^ infty U_i mid U_i text {open $ mathbb R $ and} U_i neq mathbb R text {for a finite number of $ i $ } right }. $$


For the recording it is clear whether $$ left { prod_ {i = 1} ^ nB_ {j_i} mid prod_ {i = 1} U_i in mathcal B ( mathbb R ^ infty) text {where} U_ {j_i} = B_ {j_i} text {and} U_k = mathbb R text {if $ k notin {j_1, …, j_n } $} right }, $$
is a $ sigma – $Algebra, the open set of included $ mathbb R ^ n $and thus are cylinders in $ mathcal B ( mathbb R ^ infty) $,


How can I prove that the inclusion is strict?