The conclusion is both are harmful.
My friends told me that you can not use 'is' and 'are' in the same sentence.
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When in doubt, I find it helpful to simplify the sentence. Consider these:
The conclusion is uncertain.
The conclusion is final.
The conclusion is X.
Clearly, whatever the conclusion is, it's singular and needs a singular verb.
Now let's look at what the actual conclusion is:
Both are harmful.
Again, clearly, "both" refers to two things and thus requires a plural verb, "are". You could check that by replacing both:
Cigarettes and gun battles are harmful.
Angry dragons and mean dogs are harmful.
Now, let's take that last conclusion example -- The conclusion is X. -- and replace X with what it actually is:
The conclusion is [both are harmful].
When you look at it that way, it becomes clear that the sentence as originally written is indeed correct.
My conclusion is that your friends are wrong.
You have two ideas together. One is that the conclusion is right. The other is that both A and B are harmful. You are putting these into one sentence and finding it troublesome.
You are also rushing by with spoken English and leaving out an implied word that explains things.
The conclusion is that both are purple.
It all depends on the way you define what sentence means in your context.
If you mean the string between two periods, then yes, a sentence can have an arbitrary number of conjugated verbs, and hence is and are in one sentence are allowed. For example:
The weather is fine and all people are happy.
If you consider a sentence to be the part containing subject and conjugated verb (or an infinitive sentence), then it is not possible. Then, the example above must be considered as two sentences that are contactenated.
The example in your question is really a sentence and its subclause - so it is one sentence in the first sense but two sentences in the latter sense. For better readability, there should be a that in between:
The conclusion is that both are harmful.
No reason they can't be used in the same sentence, but for the sentence you provided it's incorrect.
The context of the question is "can you use both or only one", so "both" is a solution and therefore:
"The conclusion is [the use of] both is harmful."
In the original sentence, "The conclusion is both are harmful.", this is implying that both words "is" and "are" are independently harmful, and they are not both harmful.