for G being RealNormSpace-Sequence
for seq being sequence of (product G)
for x0 being Point of (product G)
for y0 being Element of product (carr G) st x0 = y0 & seq is convergent & lim seq = x0 holds
for i being Element of dom G ex seqi being sequence of (G . i) st
( seqi is convergent & y0 . i = lim seqi & ( for m being Element of NAT ex seqm being Element of product (carr G) st
( seqm = seq . m & seqi . m = seqm . i ) ) )