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Homework Statement:
 Imagine a standard heat engine running at multiple temperature baths of temperature ##T_{max}=T_1>T_2>T_3>...>T_n=T_{min}##. It extracts heat from a certain subset of these baths and rejects heat to the others. I want to prove that a Carnot engine running between ##T_1## and ##T_n## is more efficient than this engine.
Relevant Equations:

##\eta_{Carnot} = 1\frac{T_c}{T_h}##
Conservation of energy
Let the new engine, NE, extract heat from a certain subset of these baths, and let heat obtained from the ##i^{\rm th}## bath be denoted by ##Q_i##, and let the heat rejected to the ##j^{\rm th}## be denoted by ##Q_j##. Let the engine perform an amount of work ##W##.
Now right beside this engine, I construct a reverse Carnot engine, CE, operating between ##T_1## and ##T_n## such that it takes the work NE performed. CE takes in heat ##Q_c '## from ##T_n## and releases heat ##Q_h '## to ##T_1##.
I now wrap NE and CE into a composite engine, lets call it CmE. PE operates between temperatures ##T_i## and ##T_j##, where it takes heat ##\sum _i Q_i  Q_h'## and releases heat ##\sum_j Q_j  Q_c'##.
Since $$\sum _i Q_i  Q_h ' > 0 \implies \sum _i Q_i > Q_h '$$
Dividing both sides by ##W## and inverting them,
$$\frac{W}{\sum _i Q_i} < \frac{W}{Q_h'} \implies \eta _{_{NE}} < \eta _{_{CE}}$$
However, is this a valid proof? I base this off a proof I saw about how no other engine operating between two temperatures ##T_h## and ##T_c## can be more efficient than a Carnot engine, and the proof followed the same reasoning, just without the summation symbols.
Any advice would be appreciated.
Now right beside this engine, I construct a reverse Carnot engine, CE, operating between ##T_1## and ##T_n## such that it takes the work NE performed. CE takes in heat ##Q_c '## from ##T_n## and releases heat ##Q_h '## to ##T_1##.
I now wrap NE and CE into a composite engine, lets call it CmE. PE operates between temperatures ##T_i## and ##T_j##, where it takes heat ##\sum _i Q_i  Q_h'## and releases heat ##\sum_j Q_j  Q_c'##.
Since $$\sum _i Q_i  Q_h ' > 0 \implies \sum _i Q_i > Q_h '$$
Dividing both sides by ##W## and inverting them,
$$\frac{W}{\sum _i Q_i} < \frac{W}{Q_h'} \implies \eta _{_{NE}} < \eta _{_{CE}}$$
However, is this a valid proof? I base this off a proof I saw about how no other engine operating between two temperatures ##T_h## and ##T_c## can be more efficient than a Carnot engine, and the proof followed the same reasoning, just without the summation symbols.
Any advice would be appreciated.