Utility Functions: Implying endless consumption?
When the utility function is strictly increasing, it follows that more consumption provides more utility. So it seems evident to say "more income more consumption". But discussing infinity outside mathematics makes no sense. So we want to examine

$$u'(c) &gt; 0\;\;\; \forall c,\;\;\; I\to \infty \;\;\;?\implies? \;\;\;c\to \infty$$ 

And the answer is "the question is indeterminate" or at least "it depends on the particulars", since what will matter is the rate at which Income and Consumption "go to infinity". If consumption goes faster, it will eventually deplete the infinite income, and if the consumer has infinite horizon, it will end up with no income. Etc.

If we want to entertain concepts from Non-Standard Analysis, and treat "infinity" not as a limiting concept but as something more concrete, then the answer will be the same, because this will be indeterminate also in Non-Standard Analysis, absent the particulars of the situation.