Every multitape space-S computation has a single-tape simulation in O(n+S(n)) space

Use the standard simulator from [L1] that stores the $k$ simulated tapes as marked blocks on one tape. On an input of length $n$, the first simulated tape block initially records the whole input word, contributing $n$ cells up to a constant factor. Beyond that initial block, the simulator records only cells that the multitape computation has visited, together with only constant-size bookkeeping per recorded cell and a fixed number of delimiters depending on $M$. If the simulated computation has visited at most $S(n)$ cells altogether, [L2] therefore bounds the one-tape encoding length by $O(n+S(n))$.

Therefore there exists a constant $c_M$ such that every halting run of the simulator on an input of length $n$ visits at most $c_M(n+S(n)+1)$ cells. By [L3], this is $O(n+S(n))$. Taking maxima over all length-$n$ inputs gives ${\mathrm{Space}}_{S_M}(n)=O(n+S(n))$.

The simulator still decides the same language as $M$, and step 2.1 places it in the deterministic single-tape space class described by [L4].