Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 6 / iii / b

A diamond sequence on a regular uncountable $\lambda$ implies $2^{{\aleph }_0}\le \lambda$. For every $x\subseteq \omega$, regard $x$ as a subset of $\lambda$. A correct guess above $\omega$ is exactly $x$, so the sequence's at most $\lambda$ guesses include every real.

Force with $\mathrm{Add}(\omega ,{\lambda }^{+})$ using finite conditions. The delta-system lemma and agreement on the finite root prove the countable chain condition. It preserves cardinals and the regularity of $\lambda$, while adding at least ${\lambda }^{+}$ distinct reals. The reals on distinct coordinates differ on a dense set of conditions, so their number is indeed at least this ground cardinal. In the extension,This gives the requested relative consistency over a model with the indicated regular uncountable cardinal, with the usual forcing-theoretic consistency interpretation.